Minimal Surfaces through Nevanlinna Theory 9783110989557, 9783110999822

Table of contents :
Preface
Contents
1 Some background in differential geometry
2 Analysis and geometry of complex functions
3 Minimal surfaces in R3
4 The Nevanlinna theory
5 Gauss maps of minimal surfaces in Rm
6 Nevanlinna theory on the open ball B(R0) ⊂ Cn and the nonintegrated defect relation
Bibliography
Index

Citation preview

Min Ru Minimal Surfaces through Nevanlinna Theory

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu, Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L. Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany

Volume 92

Min Ru

Minimal Surfaces through Nevanlinna Theory �

Mathematics Subject Classification 2020 32H30, 52A10 Author Prof. Min Ru University of Houston Department of Mathematics Houston, TX 77204 USA [email protected]

ISBN 978-3-11-099982-2 e-ISBN (PDF) 978-3-11-098955-7 e-ISBN (EPUB) 978-3-11-098996-0 ISSN 0179-0986 Library of Congress Control Number: 2023931747 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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Dedicated to my lovely granddaughter Riley Ru

Preface In the mid-1960s, Chern and Osserman [20] began a program to study the value distribution properties of the Gauss maps of minimal surfaces immersed in ℝm through the theory of holomorphic curves (Nevanlinna theory). This program proved crucial when Fujimoto [27], solved the long-standing “five-point theorem” Conjecture in 1988, which states that the classical Gauss map of a nonflat complete minimal surface immersed in ℝ3 can omit at most four distinct values. Before that, Xavier [88] already obtained a weaker “seven-point theorem” by using the logarithmic derivative lemma in the Nevanlinna theory. Since then, there have been many important developments in this area and this book describes the progress of this program. The theory of minimal surfaces is naturally linked to the complex analysis. Under isothermal coordinates, a surface is minimal if and only if its parameterizations are harmonic. By considering the minimal surface as a complex Riemann surface, the Weierstrass–Enneper representation associates it with a pair of functions f , g, where f is analytic and g is meromorphic. The function g is globally defined, and by regarding g as the map g : M → ℙ1 (ℂ), it becomes the Gauss map of the minimal surface M after we identify the unit sphere S 2 with ℙ1 (ℂ) using the stereographic projection. In this way, we can apply the theory of complex analysis, particularly the Nevanlinna theory, to study minimal surfaces. This book is intended to be self-contained, with few prerequisites making it suitable for undergraduate students as well as graduate students and researchers. Chapters 1 and 2 introduce the mathematics necessary to investigate minimal surfaces, including the basics of differential geometry and complex analysis. Chapter 2 also includes the theory of algebraic curves, serving as an introduction to the theory of holomorphic curves (Nevanlinna theory) and playing a crucial rule in the study of Gauss maps of the complete minimal surfaces with finite total curvature. Chapter 3 focuses the heart of the book, which is the the value distribution of the Gauss maps of minimal surfaces immersed in ℝ3 . Chapter 4 gives a detailed presentation of the Nevanlinna theory and Chapter 5 studies the Gauss maps of minimal surfaces immersed ℝm . Finally in Chapter 6, we establish the nonintegrated defect relation for the meromorphic maps of complete Kähler manifolds whose universal cover is the ball B(R0 ) (0 < R0 ≤ ∞) into ℙN (ℂ) following Xaiver’s method developed in his proof of the “seven-point theorem”. I would like to acknowledge the financial support of the U. S. National Science Foundation (through grants DMS-9596181, DMS-9800361) and U.S. National Security Agency (through grants MDA904-99-1-0034, MDA904-01-1-0051, MDA904-03-1-0067, H98230-051-0042, H98230-07-1-0050, H98230-09-1-0004, H98230-11-1-0201), as well as the Simons Foundation Mathematics and Physical Sciences-Collaboration Grants for Mathematicians (award numbers 521604 and 916140). I am grateful to my students Qili Cai and Chin

https://doi.org/10.1515/9783110989557-201

VIII � Preface Jui Yang for reading a preliminary version with corrections and suggestions. Lastly, I would like to thank my wife Yu Shen, my son Aaron Ru, and my daughter Christina Ru for their love, support, understanding, and patience. During the time the book was written, my granddaughter Riley Ru was born, and I dedicate this book to her.

Contents Preface � VII 1 1.1 1.2 1.3 1.4

Some background in differential geometry � 1 Curves in ℝn � 1 Surfaces in ℝ3 and ℝn � 2 The Gauss–Bonnet theorem � 9 The abstract surfaces � 15

2 2.1 2.2 2.3 2.4 2.5

Analysis and geometry of complex functions � 19 Holomorphic functions � 19 Harmonic functions � 24 The Ahlfors–Schwarz lemma and the negative curvature method � 25 The Riemann surfaces � 35 The theory of algebraic curves � 38

3 3.1 3.2 3.3 3.4

Minimal surfaces in ℝ3 � 47 The minimal surfaces � 47 The Weierstrass–Enneper representation � 50 Gauss maps of complete minimal surfaces in ℝ3 � 58 Minimal surfaces with finite total curvatures � 70

4 4.1 4.2 4.3 4.4

The Nevanlinna theory � 74 Nevanlinna theory of meromorphic functions � 74 Cartan’s second main theorem for holomorphic curves � 89 The degenerate case � 112 Holomorphic curves through Ahlfors’ negative curved method � 121

5 5.1 5.2

Gauss maps of minimal surfaces in ℝm � 135 Minimal surfaces in ℝm � 135 Value distributions of the (generalized) Gauss maps of complete minimal surfaces in ℝm � 140 The Gauss curvature estimate � 145 Gauss maps of minimal surfaces in ℝm with finite total curvature � 155

5.3 5.4 6 6.1 6.2 6.3

Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation � 160 Meromorphic functions on B(R0 ) ⊂ ℂn � 160 Nevanlinna theory on B(R0 ) ⊂ ℂn � 167 The nonintegrated defect relation � 174

X � Contents Bibliography � 189 Index � 193

1 Some background in differential geometry The goal of this chapter is developing some basics of differential geometry needed to investigate minimal surfaces in ℝ3 and ℝn .

1.1 Curves in ℝn We begin with some basic notions. A parameterized curve in ℝn is a map α : I → ℝn , where I is an interval in ℝ. It is said to be regular if α′ (t) ≠ 0 for all t ∈ I. We can write α(t) = (x1 (t), x2 (t), . . . , xn (t)). For example, α(t) = (cos t, sin t, t) for t ∈ ℝ is a parameterization of a helix. The tangent vector (or velocity vector) of α is α′ (t) = (x1′ (t), . . . , xn′ (t)). The arc length of a curve on [a, b] is b

󵄩 󵄩 length(α) = ∫󵄩󵄩󵄩α′ (t)󵄩󵄩󵄩dt. a

A curve α(t) is of unit speed if ‖α′ (t)‖ ≡ 1. Any nonunit speed regular curve α(t) can be reparameterized by changing the parameter t to the arc-length parameter s, i. e., s(t) = t ∫a ‖α′ (u)‖du, to form a unit speed curve β(s) := α(t(s)), where t = t(s) is the inverse of the function s = s(t). Therefore we can always assume that the curves we will be discussing are unit-speed curves. This assumption means that we are only interested in the geometric shape of a regular curve since reparameterizing does not change the shape of a curve. Let α : I → ℝ3 be a unit-speed curve. We are interested in the rate of change of the tangent vector. Denote by T(s) := α′ (s) the unit-tangent vector. Then the curvature of α is 󵄩 󵄩 󵄩 󵄩 κ(s) := 󵄩󵄩󵄩T′ (s)󵄩󵄩󵄩 = 󵄩󵄩󵄩α′′ (s)󵄩󵄩󵄩. Denote N(s) :=

T′ (s) , ‖T′ (s)‖

the principal normal of α, and B(s) = T(s) × N(s), the binormal

of α. We have the following Frenet formulas: T′ = κN,

N′ = −κT + τB,

B′ = −τN,

where τ(s) given by B′ (s) = −τ(s)N(s) is the torsion of α. Example. What is the curvature of a circle at point p? The location of p does not matter because of the symmetry of the circle. A circle of radius r can be parameterized by α(t) = (r cos t, r sin t, 0), 0 < t < 2π. Notice that this is not a unit-speed curve since α′ (t) = (−r sin t, r cos t, 0), and hence https://doi.org/10.1515/9783110989557-001

2 � 1 Some background in differential geometry ‖α′ (t)‖ = r. So we reparameterize it into a unit-speed curve by letting β(s) = (r cos(s/r), r sin(s/r), 0), 0 < s < 2πr. Therefore

s r

= t to get

β′ (s) = (− sin(s/r), cos(s/r), 0),

1 1 β′′ (s) = (− cos(s/r), − sin(s/r), 0). r r Thus 󵄩 󵄩 1 κ(s) = 󵄩󵄩󵄩β′′ (s)󵄩󵄩󵄩 = , r that is, the curvature of a circle of radius r is r1 . Notice that if r is large, then the curvature is small, whereas if r is small, then the curvature is large.

1.2 Surfaces in ℝ3 and ℝn A (local) parameterized surface in ℝn is a one-to-one map x : U ⊂ ℝ2 → ℝn , where U is an open subset of ℝ2 . In this book, we always assume that x is of class at least C 2 , that is, the partial derivatives of x up to the second order all exist and are continuous. We can write x(u, v) = (x1 (u, v), x2 (u, v), . . . , xn (u, v)), (u, v) ∈ U. The surface is said to be regular if xu × xv ≠ 0 for all points (u, v) ∈ U. A global surface in ℝn is a set M such that for every point p ∈ M, there is a local parameterization x : U ⊂ ℝ2 → M ⊂ ℝn with p ∈ x(U). We now focus on the surfaces M in ℝ3 and introduce various notions of curvatures for M. A curve on a surface M is a mapping from an interval to the surface, that is, α : I ⊂ ℝ → M ⊂ ℝ3 . We can write α(t) = x(u(t), v(t)) for some functions u, v : I → ℝ. Given p ∈ M, a tangent vector to M at p ∈ M is the tangent vector of some curve α : (−ϵ, ϵ) → M (for some ϵ > 0) with α(0) = p, that is, it is α′ (0). The tangent space of M at p ∈ M, denoted by Tp M, consists of all of the tangent vectors to M at p ∈ M, that is, Tp M = {α′ (0) | α : (−ϵ, ϵ) → M for some ϵ > 0 with α(0) = p}, which is a vector space with standard basis {xu |p , xv |p }. The unit normal to the surface n is given by (n ∘ x) =

xu × xv . ‖xu × xv ‖

The first fundamental form (also called a metric) I assigns to every p ∈ M the map Ip : Tp M × Tp M → ℝ given by Ip (v, w) = v ⋅ w for all v, w ∈ Tp M, where v ⋅ w is the standard dot product of v, w (as vectors in ℝ3 ). If we write v = axu + bxv and w = cxu + dxv , then I(v, w) = v ⋅ w = (ac)g11 + (ad + bc)g12 + (bd)g22 , where g11 = xu ⋅ xu , g12 = g21 = xu ⋅ xv , g22 = xv ⋅ xv . For this reason, we call g11 , g12 , g21 , g22 the coefficients of the first fundamental form.

1.2 Surfaces in ℝ3 and ℝn

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We now ready to introduce various notions of the curvatures of surfaces. As we have seen above, the definition of the curvature for curves used the (unit) tangent vector of the curves. However, in the surface case, dim Tp M = 2, so the choices of the tangent vectors for M are not canonical. Instead, we use its unit-normal n to define the curvatures. We use two ways to introduce various notions of curvatures for surfaces: (1) through curves on the surfaces and (2) through the shape operator Sp := −dn|p . We first discuss the approach of using curves on the given surfaces. The normal curvature of curves in a surface Let α be a unit-speed curve on a surface M. We introduce the Darboux frame along the curve, {α′ (t), n(t), n(t) × α′ (t)}, where n(t) is the unit normal to the surface restricted to the α(t) (i. e., n(t) = n(α(t))). It is a moving orthonormal frame, so every other vector can be written as a linear combination of the vectors in the Darboux frame. Notice that α′′ is perpendicular to α′ , so we can write α′′ (t) = κn (t)n(t) + κg (t)(n(t) × α′ (t)), where κn (t) = α′′ (t) ⋅ n(t), κg (t) = α′′ (t) ⋅ (n(t) × α′ (t)). The scalars κn (t) and κg (t) are called the normal curvature and the geodesic curvature of α at the point p = α(t). We have κ2 (t) = κn2 (t) + κg2 (t), where κ(t) is the curvature of α. The normal curvature of surfaces with respect to a given tangent vector v ∈ Tp M Let M be a surface in ℝ3 , p ∈ M, and let v ∈ Tp M be a unit tangent vector (direction). Then there is a (unit-speed) curve α : (0, ϵ) → M such that α(t0 ) = p and α′ (t0 ) = v for some t0 ∈ (0, ϵ). We define the normal curvature of M at p in the direction v, denoted by κn,v (p), as the normal curvature of the curve α at p, that is, κn,v (p) := α′′ (t0 ) ⋅ n(p), where n(p) is the unit normal at the point p. The normal curvature depends on the directions. The max κn,v (p) and min κn,v (p), where max and min are taken over all v ∈ Tp M with ‖v‖ = 1, are called the principal curvatures. Note that the above definition uses the curves in a surface. Theorem 1.2.1 below tells us that κn,v (p) is also related to the shape operator Sp . The Gauss map and shape operator There is a unique, up to a direction, unit normal n(p) to the surface M at p ∈ M, which is normal to the tangent plane Tp M. We fix a direction of n (in this case, we say that M is oriented, which we always assume). So v ∈ Tp M if and only if v ⋅ n(p) = 0, that is, v is perpendicular to n(p). The Gauss map of the surface M is the map n : M → S 2 ⊂ ℝ3 that

4 � 1 Some background in differential geometry sends every point p ∈ M to the unit normal n(p) to M at the point p. Define the shape operator Sp = −dn|p . A key observation about Sp is that, in fact, Sp (v) is inside Tp M for every v ∈ Tp M, and it is a symmetric linear operator, that is, Sp (v) ⋅ w = v ⋅ Sp (w) for all v, w ∈ Tp M. The second fundamental form of M, denoted by II, assigns to each p ∈ M the map IIp : Tp M × Tp M → ℝ given by IIp (v, w) = w ⋅ Sp (v) for v, w ∈ Tp M. Note that Sp (xu ) = −dn(xu ) = −nu and Sp (xv ) = −nv , and hence under the standard basis {xu , xv }, IIp (axu + bxv , cxu + dxv ) = (ac)h11 + h12 (bc + ad) + h22 (bd), where h11 = IIp (xu , xu ), h12 = h21 = IIp (xu , xv ), h22 = IIp (xv , xv ). We call the data {h11 , h12 , h21 , h22 } the coefficients of the second fundamental form. Note that, by the definition, h11 = xu ⋅ Sp (xu ) = −xu ⋅ nu = n ⋅ xuu , where in the last equation we used the fact that xu ⋅ n = 0, so (xu ⋅ n)u = 0, which gives −xu ⋅ nu = n ⋅ xuu . Similarly, h12 = −nu ⋅ xv = n ⋅ xuv and h22 = −xv ⋅ nv = n ⋅ xvv . The following theorem links the shape operator with the normal curvature. Theorem 1.2.1. κn,v (p) = Sp (v) ⋅ v. Proof. Let x : U → M be a parameterization of M. Take a (unit-speed) curve α : (0, ϵ) → M on M with α(t0 ) = p and α′ (t0 ) = v for 0 < t0 < ϵ. Write α(t) = x(u(t), v(t)). Then α′ (t) = u′ (t)xu (t)+v′ (t)xv (t), where xu (t) = xu (u(t), u(t)) and xv (t) = xv (u(t), u(t)). Hence α′′ (t) = u′ (t)2 xuu (t) + 2u′ (t)v′ (t)xuv (t) + v′ (t)2 xvv (t) + u′′ (t)xu (t) + v′′ (t)xv (t).

Therefore κn,v (p) = α′′ (t0 ) ⋅ n(p)

= u′ (t0 )2 (xuu (t0 ) ⋅ n(p)) + 2u′ (t0 )v′ (t0 )(xuv (t0 ) ⋅ n(p)) + v′ (t0 )2 (xvv (t0 ) ⋅ n(p))

= h11 (p)u′ (t0 )2 + 2h12 (p)u′ (t0 )v′ (t0 ) + h22 (p)v′ (t0 )2 = IIp (α′ (t0 ), α′ (t0 )) = IIp (v, v) = Sp (v) ⋅ v.

1.2 Surfaces in ℝ3 and ℝn

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Since Sp is a symmetric linear transformation, from the linear algebra we know that real eigenvalues of Sp always exist. Let κ1 and κ2 be eigenvalues of Sp with unit eigenvectors v1 , v2 ∈ Tp M, that is, Sp (v1 ) = κ1 v1 and Sp (v2 ) = κ2 v2 . Then κn,v1 (p) = Sp (v1 ) ⋅ v1 = κ1 ‖v1 ‖2 = κ1 , and, similarly, κn,v2 (p) = κ2 , that is, κ1 and κ2 are the (special) normal curvatures of M at p ∈ M in the directions of v1 and v2 , respectively. They are indeed (see Euler’s formula below) the maximal and minimal curvatures among all normal curvatures of M at p. Therefore the eigenvalues κ1 and κ2 of Sp are the principal curvatures of M at p. We call the corresponding unit eigenvectors the principal directions. Note that if κ1 ≠ κ2 , then v1 and v2 are orthogonal. Theorem 1.2.2 (Euler’s formula). Let e1 and e2 be the unit vectors in the principal directions at p corresponding to the principal curvatures κ1 and κ2 . Suppose that v = cos θe1 + sin θe2 for some θ ∈ [0, 2π). Then κn,v = κ1 cos2 θ + κ2 sin2 θ. Proof. By Theorem 1.2.1 we have κn,v = Sp (v) ⋅ v

= (Sp (cos θe1 + sin θe2 )) ⋅ (cos θe1 + sin θe2 )

= cos2 θSp (e1 ) ⋅ e1 + 2 cos θ sin θSp (e1 ) ⋅ e2 + sin2 θSp (e2 ) ⋅ e2 = cos2 θκ1 e1 ⋅ e1 + 2 cos θ sin θκ1 e1 ⋅ e2 + sin2 θκ2 e2 ⋅ e2 = κ1 cos2 θ + κ2 sin2 θ,

where we used the fact that Sp (ei ) = κi ei , i = 1, 2, and e1 ⋅ e2 = 0 (see the remark above), and ‖ei ‖ = 1, i = 1, 2. This finishes the proof. The mean curvature and Gauss curvature for surfaces in ℝ3 Let M be a surface in ℝ3 , and let p ∈ M. Let κ1 and κ2 be the principal curvatures of M at p. We define the Gaussian curvature of M at p as K(p) = κ1 κ2 and the mean curvature of M at p as H(p) =

κ1 + κ2 . 2

6 � 1 Some background in differential geometry Since κ1 and κ2 are eigenvalues of Sp , we have K(p) = κ1 κ2 = det(Sp ) and H(p) =

κ1 + κ2 1 = trace(Sp ). 2 2

We have K=

det(hij ) det(gij )

=

2 h11 h22 − h12 , 2 g11 g22 − g12

H=

h11 g22 − 2h12 g12 + h22 g11 . 2) 2(g11 g22 − g12

(1.1)

The following theorem gives a geometric interpretation of the Gauss curvature K. Theorem 1.2.3. Let p ∈ M with K(p) ≠ 0. Then K(p) = lim

Ω→p

Area n(Ω) Area Ω

for any Ω containing and shrinking to p. Proof. The area of Ω is ∬ ‖xu × xv ‖dudv, Ω

and the area of n(Ω) is 󵄩 󵄩 ∬󵄩󵄩󵄩dn(xu ) × dn(xv )󵄩󵄩󵄩dudv. Ω

It is easy to check that dn(xu ) × dn(xv ) = K(xu × xv ). This gives us the result. The Gauss curvature is intrinsic We prove the Gauss theorem egregium: The Gauss curvature is intrinsic, that is, it only depends on the metric (gij ). To do so, let x(u1 , u2 ) be a parameterization of a surface M. We write x1 := xu1 , x2 := xu2 , xi1 :=

𝜕xi , 𝜕u1

xi2 := 2

𝜕xi 𝜕u2

for i = 1.2. Write

xij = ∑ Γkij xk + hij n, k=1

(1.2)

1.2 Surfaces in ℝ3 and ℝn

�

7

where Γkij , 1 ≤ i, j, k ≤ 2, are called the Christoffel symbols, and hij = n ⋅ xij are the coefficients of the second fundamental form. Note that the Christoffel symbols are symmetric with respect to the lower indices: Γkij = Γkji

for 1 ≤ i, j, k ≤ 2.

Let gij := xi ⋅ xj be the first fundamental form (also called the metric). We first derive the formula Γkij =

𝜕gjt 𝜕gij 𝜕g 1 ∑ g kt ( itj + i − t ), 2 𝜕u 𝜕u 𝜕u

(1.3)

where (g kt ) is the inverse matrix of (gij ). Indeed, from the above expression of xij in (1.2), taking the dot product with xl (notice that xl ⋅ n = 0), we get, for 1 ≤ i, j, l ≤ 2, 2

xij ⋅ xl = ∑ Γkij gkl . k=1

Therefore 2 2 𝜕git 𝜕(xi ⋅ xt ) k = = x ⋅ x + x ⋅ x = Γ g + Γktj gki . ∑ ∑ ij t i tj kt ij 𝜕uj 𝜕uj k=1 k=1

Cyclical permutation of the indices, recalling that Γkij = Γkji , yields 𝜕gjt 𝜕ui

𝜕gij 𝜕ut

2

2

k=1

k=1

= ∑ Γkij gkt + ∑ Γkti gkj , 2

2

= ∑ Γkti gkj + ∑ Γktj gki . k=1

k=1

Hence 2 𝜕git 𝜕gjt 𝜕gij + − = 2 Γkij gkt . ∑ 𝜕uj 𝜕ui 𝜕ut k=1

Taking the inverse matrix of (gij ) proves identity (1.3). From this we see that Γkij only depends on the metric gij , so it is an intrinsic quantity. Let Γm ij,k :=

𝜕Γm ij 𝜕uk

.

Write, for 1 ≤ i, j, k, m ≤ 2, the Riemann curvature tensor of the first kind

8 � 1 Some background in differential geometry 2

m m l m l m Rm ijk := Γik,j − Γjk,i + ∑(Γik Γjl − Γjk Γil ), l=1

(1.4)

and define the curvature tensor of the second kind 2

Rijkl := ∑ Rm ijk gml . m=1

Theorem 1.2.4 (Gauss’s theorem egregium). R1212 . det(gij )

K=

(1.5)

Hence K is intrinsic. Proof. We use the fact that (xij )k = (xik )j . Step 1: We first compute (xij )k . We start from 2

xij = ∑ Γlij xl + hij n.

(1.6)

l=1

Hence (xij )k =

𝜕xij 𝜕uk 2

2

= (∑ Γlij xl + hij n) l=1

k

2

= ∑ Γlij,k xl + ∑ Γlij xlk + hij,k n + hij nk . l=1

l=1

(1.7)

On the other hand, it is easy to derive that the matrix of Sp with respect to the standard basis {x1 , x2 } is ℱI−1 ℱII : 2

Sp (xi ) = ∑ hij g jk xk , j,k=1

where (g ik ) is the inverse matrix of (gij ). Since ni :=

𝜕n = dn(xi ) = −Sp (xi ), 𝜕ui

we get 2

ni = − ∑ hij g jk xk . j,k=1

Thus by combining (1.6), (1.7), and (1.8) we get

(1.8)

1.3 The Gauss–Bonnet theorem

2

2

2

2

m=1

l=1

l=1

l=1

� 9

l m lm l (xij )k = ∑ (Γm ij,k + ∑ Γij Γlk − ∑ hij hkl g )xm + (hij,k + ∑ Γij hlk )n.

If we write (xij )k = ∑2m=1 Am ijk xm + Bijk n, then 2

2

l=1

l=1

m l m lm Am ijk = Γij,k + ∑ Γij Γlk − ∑ hij hkl g . m Step 2: Using the fact that (xij )k = (xkj )i , we get Am ijk = Akji . This gives 2

2

l=1

l=1

m l m l m lm Γm ij,k − Γkj,i + ∑(Γij Γlk − Γkj Γli ) = ∑(hij hkl − hkj hil )g .

Recalling that Γkij = Γkij , by definition (1.4) the last equation becomes 2

lm Rm ikj = ∑(hij hkl − hkj hil )g . l=1

Thus 2

hij hkl − hkj hil = ∑ glm Rm ikj = Rikjl . m=1

Therefore K=

2 h11 h22 − h12 R1212 = , 2 det(gij ) g11 g22 − g12

which finishes the proof. See also [79, 12] for alternative proofs.

1.3 The Gauss–Bonnet theorem The Gauss–Bonnet theorem is a fundamental tool in the study of minimal surfaces, especially in the case where the surface is of finite total curvature. It is also a foundation of the theory of Riemann surfaces and the Nevanlinna theory. In this section, we prove the Gauss–Bonnet theorem. We first give a new proof of the Gauss theorem egregium using the moving frame method. According to the Gram–Schmidt process in linear algebra, we can always conx x vert {xu , xv } into an orthonormal basis {e1 , e2 }. Indeed, we can take e1 = ‖xu ‖ = √gu and u

e2 =

xv − (xv ⋅ e1 )e1 g 1 = (√g11 xv − 12 xu ). ‖xv − (xv ⋅ e1 )e1 ‖ √g g − g 2 √g11 11 22 12

11

10 � 1 Some background in differential geometry Then {e1 , e2 , e3 }, where e3 = n, is an orthonormal moving frame for ℝ3 . Write e1u = b111 e1 + b211 e2 + c11 e3

e1v = b112 e1 + b212 e2 + c12 e3

e2u = b121 e1 + b221 e2 + c21 e3

e2v = b122 e1 + b222 e2 + c22 e3 ,

where bij , 1 ≤ i, j ≤ 2, are functions on M. From the first equation above, b111 = e1u ⋅ e1 . Since e1 ⋅ e1 = 1, by taking the partial with respect to u we get 2e1u ⋅ e1 = 0. Hence e1u ⋅ e1 = 0, i. e. b11 = 0. In a similar way, we can get b112 = b221 = b222 = 0. By differentiating e1 ⋅ e2 = 0 we get e1u ⋅ e2 = −e1 ⋅ e2u , that is, b211 = −b121 ; similarly, we have b212 = −b122 . Thus the above equations become e1u = b211 e2 + c11 e3

e1v = b212 e2 + c12 e3

e2u = −b211 e1 + c21 e3

e2v = −b212 e1 + c22 e3 .

(1.9)

We claim that K=

1 2 √g11 g22 − g12

(c11 c22 − c12 c21 ).

(1.10)

Indeed, from e1 ⋅ e3 = 0, differentiating with respect to u on both sides, we get e1u ⋅ e3 + e1 ⋅ e3u = 0, so c11 = e1u ⋅ e3 = −e1 ⋅ e3u . Similarly, c12 = −e1 ⋅ e3v , c21 = −e2 ⋅ e3u , and c22 = −e2 ⋅ e3v . Hence, also by noticing that e3u ⋅ e3 = e3v ⋅ e3 = 0, e3u = −c11 e1 − c21 e2

e3v = −c12 e1 − c22 e2 .

2 e + g e , we have Since xu = √g11 e1 and √g11 xv = √g11 g22 − g12 2 12 1

e3u = nu = −Sp (xu ) = −√g11 Sp (e1 ) and 2 S (e ) − g S (e ). √g11 e3v = √g11 nv = −√g11 Sp (xv ) = −√g11 g22 − g12 p 2 12 p 1

Therefore we get Sp (e1 ) = −

1 1 e3u = (c11 e1 + c21 e2 ), √g11 √g11

1.3 The Gauss–Bonnet theorem

Sp (e2 ) = = =

1 2 √g11 g22 − g12

√g11 √g11 g22 −

2 g12

√g11

2 √g11 g22 − g12

� 11

(−√g11 e3v − g12 Sp (e1 )) (c12 e1 + c22 e2 − ((c12 −

g12 (c e + c21 e2 )) g11 11 1

g12 g c11 )e1 + (c22 − 12 c21 )e2 ). g11 g11

So the matrix of the shape operator Sp with respect to the basis γ = {e1 , e2 } is 1 c √g11 11

[Sp ]γ = (

1 c √g11 21

√g11

(c12 −

g12 c ) g11 11

√g11

(c22 −

g12 c ) g11 21

2 √g11 g22 −g12 2 √g11 g22 −g12

).

Thus K = det(Sp ) = =

1

1 √g11 g22 −

2 √g11 g22 − g12

2 g12

(c11 (c22 −

g g12 c ) − c21 (c12 − 12 c11 )) g11 21 g11

(c11 c22 − c12 c21 ).

This proves the claim. Using equations (1.9), it is easy to check that e1u ⋅ e2v − e2u ⋅ e1v = c11 c22 − c12 c21 . Hence (b211 )v − (b212 )u = −(e2u ⋅ e1 )v + (e2v ⋅ e1 )u

= −(e2 )uv ⋅ e1 − e2u ⋅ e1v + (e2 )uv ⋅ e1 + e2v ⋅ e1u = e1u ⋅ e2v − e2u ⋅ e1v = c11 c22 − c12 c21 .

Thus we obtain the formula for the Gauss curvature K=

(b211 )v − (b212 )u 2 √g11 g22 − g12

,

(1.11)

which shows that K is intrinsic. We now derive the formula for computing the geodesic curvature. Let α(t) be a unit-speed curve in M. Let ei (t) = (ei ∘ α)(t) for i = 1, 2. Then we can write α′ (t) = cos θ(t)e1 (t) + sin θ(t)e2 (t), where θ(t) is the angle from e1 (t) to α′ (t). Notice that the vector n(t) × α′ (t) is obtained by rotating the vector α′ (t) counterclockwise 90 degrees, and hence

12 � 1 Some background in differential geometry π π )e (t) + sin(θ(t) + )e2 (t) 2 1 2 = − sin θ(t)e1 (t) + cos θ(t)e2 (t).

n(t) × α′ (t) = cos(θ(t) +

Also, α′′ (t) = −θ′ (t) sin θ(t)e1 (t) + θ′ (t) cos θ(t)e2 (t) + cos θ(t)e′1 (t) + sin θ(t)e′2 (t).

Thus, by the geodesic curvature formula for α, κg (t) = α′′ (t) ⋅ (n(t) × α′ (t))

= (ϕ12 (t) + θ′ (t))(cos2 θ(t) + sin2 θ(t)) = ϕ12 (t) + θ′ (t),

where ϕ12 (t) = e′1 (t) ⋅ e2 (t). So we derived the Liouville formula κg (t) = ϕ12 (t) + θ′ (t), where θ(t) is the angle from e1 (t) to α′ (t). We now ready to prove the following Gauss–Bonnet formula. Theorem 1.3.1 (Local Gauss–Bonnet theorem for the regions with smooth boundary). Let M be a parameterized surface given by x : U → ℝ3 . Let α be a unit-speed simple smooth closed curve in M, and let R be the interior region bounded by the curve α. Then L

∬ KdA + ∫ κg ds = 2π, R

0

where L is the total arc length of α, K is the Gauss curvature, and dA is the element of area 2 dudv. given by dA = √g11 g22 − g12 Proof. By (1.11), K =

(b211 )v −(b212 )u 2 √g11 g22 −g12

. Recall the Green theorem in calculus: for any smooth

functions P(u, v), Q(u, v) defined around the region D ⊂ ℝ2 , ∬( D

𝜕Q 𝜕P − )dudv = ∫ (Pdu + Qdv). 𝜕u 𝜕v 𝜕D

Thus, for D ⊂ ℝ2 with x(D) = R, 2 dudv = ∬((b2 ) − (b2 ) )dudv ∬ KdA = ∬ K √g11 g22 − g12 11 v 12 u R

D

D

1.3 The Gauss–Bonnet theorem

=−

∫ (b211 du

+

� 13

L

b212 dv)

= − ∫((e1u ⋅ e2 )u′ (s) + (e1v ⋅ e2 )v′ (s))ds 0

𝜕D L

L

= − ∫(e′1 (s) ⋅ e2 (s))ds = − ∫ ϕ12 (s)ds. 0

0

On the other hand, by the Liouville formula, L

L

L

L

∫ κg ds = ∫(ϕ12 (s) + θ (s))ds = ∫ ϕ12 (s)ds + θ(L) − ϕ(0) = ∫ ϕ12 (s)ds + 2π, ′

0

0

0

0

where we used the fact that 𝜕R is a smooth closed curve, so θ(L)−ϕ(0) = 2π. This finishes the proof. We now consider the case that 𝜕R is piecewise smooth but not entirely smooth. Theorem 1.3.2 (Local Gauss–Bonnet). Let M be a parameterized surface given by x : U → ℝ3 . Suppose that R ⊂ M is a simply connected region. If the curve 𝜕R has exterior angles ϵj , j = 1, . . . , q, then q

∫ κg ds + ∬ KdA + ∑ ϵj = 2π. j=1

R

𝜕R

Proof. If 𝜕R is smooth, then △θ = 2π, so 2π = △θ = ∫ κg ds + ∬ KdA. 𝜕R

R q

When 𝜕R has corners, the unit tangent vector turns less by the amount ∑j=1 ϵj , and so the result follows. This finishes the proof. When R = T is a triangle on M (i. e., a region with boundary consisting of three smooth curves) with interior angels A, B, C, it implies (with ϵ1 = π − A, ϵ2 = π − B, and ϵ3 = π − C) the following: Theorem 1.3.3 (Gauss formula for an embedded triangle). Let M be a parameterized surface given by x : U → ℝ3 . Let T be an embedded triangle on M with interior angels A, B, C. Then ∬ KdA + ∫ κg ds = A + B + C − π. T

𝜕T

We now derive the global version of the Gauss–Bonnet theorem. Consider an oriented surface with piecewise smooth boundary. T. Rado proved in 1925 that any such surface M can be triangulated, that is, we may write M = ⋃m λ=1 △λ , where

14 � 1 Some background in differential geometry (i) △λ is the image of a triangle under a (oriented-preserving) parameterization; (ii) △λ ∩ △ν is either an empty or single vertex, or a single edge; (iii) when △λ ∩ △ν consists of a single edge, the orientations of the edge are opposite in △λ and △ν ; and (iv) at most one edge of △λ is contained in the boundary of M. Definition 1.3.4. Given a triangulation 𝒯 of a surface M with V vertices, E edges, and F faces, we define the Euler characteristic χ(M, 𝒯 ) = V − E + F. It is independent of the choice of the triangulation 𝒯 , so we just denote it by χ(M). We can triangulate a disk as follows:

Theorem 1.3.5 (Gauss–Bonnet theorem). Let M be a compact surface in ℝ3 without boundary. Then ∬ KdA = 2πχ(M), M

where K is the Gauss curvature, dA is the area measure, and χ(M) is the Euler characteristic of M. Proof. Let M = ⋃ △λ be a triangulation. Then ∬ KdA = ∑ ∬ KdA. M

λ △ λ

Using the local Gauss–Bonnet theorem for triangles △λ , we get

1.4 The abstract surfaces

� 15

3

∬ KdA + ∫ κg ds = ∑ ℓj − π, △λ

j=1

𝜕△λ

where ℓj , 1 ≤ j ≤ 3, are the three interior angles of the triangle △λ . Since the integrals ∫𝜕△ κg ds cancel in pairs due to the opposite orientations, we have λ

3

∬ KdA = ∑ ∑ ℓj − πF, λ j=1

M

where F is the number of triangles △λ (i. e., the number of faces). Notice that at each vertex, the sum of all interior angles is 2π, so ∬ KdA = 2πV − πF, M

where V = # of vertices. Use the fact that M does not have boundary, every triangle has three edges, and each edge is shared by two triangles, and hence 3F = 2E, so ∬ KdA = 2πV − πF = 2πV − π(2F − 2E) M

= 2π(E + V − F) = 2πχ(M).

This proves our theorem. A typical compact surface is the unit sphere (with no holes), a torus, or a torus with g holes. The number of holes of M, which is also called the genus of M, denoted by g, is a topological invariant property (i. e., g does not change if we continuously change M, as long as we do not break it down). We can in fact show that χ(M) = 2 − 2g.

1.4 The abstract surfaces Recall that a parameterization is a one-to-one regular mapping x : D ⊂ ℝ2 → ℝm of an open set D. A surface in ℝm is a subset M ⊂ ℝm such that each point of M has a neighborhood (in M) contained in the image of some parameterization x : D → M ⊂ ℝm ,

where D ⊂ ℝ2 .

To define a two-dimensional abstract surface (or a general n-dimensional manifold), we need to remove ℝm in the definition. The following are important ingredients we want to keep: (i) M = set with concept of continuity; (ii) For every p ∈ M, there is subset U of M such that p ∈ U and we can assign coordinates on U; (iii) Change of coordinates is differentiable (or smooth).

16 � 1 Some background in differential geometry What do we mean by coordinates (in the two-dimensional case)? Coordinate is a way to associate every object (point) we want to study with an ordered tuple of real numbers. For example, under the Cartesian coordinate system, every point on a plane can be associated with a pair of numbers. It is very important, since all mathematics are expressed in numbers. In our situation the object we want to study is an n-dimensional manifold M, that is, we want to associate every point p with an ordered tuple of real numbers. Also note that for most of times, it suffices to only work on it locally. For example, for the (local) surface x : D ⊂ ℝ2 → M ⊂ ℝ3 , we can associate (locally, i. e., at every point in x(D) ⊂ M) every point p with x−1 (p) ∈ ℝ2 . So the inverse x−1 is called the coordinate map. The definition of n-manifolds can be defined in terms of coordinate systems (coordinate maps). Definition 1.4.1. An n-dimensional differentiable (reps., smooth) manifold is a set M together with a family of one-to-one maps ϕα : Uα ⊂ M → ℝn , α ∈ Λ, with ϕα (Uα ) being open in ℝn into M, such that (1) ⋃α∈Λ Uα = M, and (2) For each pair α, β with Uα ∩ Uβ ≠ 0, the maps n n ϕβ ∘ ϕ−1 α : ϕα (Uα ∩ Uβ ) ⊂ ℝ → ϕβ (Uα ∩ Uβ ) ⊂ ℝ

and n n ϕα ∘ ϕ−1 β : ϕβ (Uα ∩ Uβ ) ⊂ ℝ → ϕα (Uα ∩ Uβ ) ⊂ ℝ

are differentiable (smooth). The map ϕα : Uα ⊂ M → ℝn is called the coordinate map, and its inverse ϕ−1 α : ϕα (Uα ) → M is called a local parameterization of M, similarly to our previous discussion in the surface case. An abstract surface is a two-dimensional differentiable manifold. Example. Consider S 2 = {(x, y, z) | x 2 + y2 + z2 = 1} ⊂ ℝ3 . Let N = (0, 0, 1) be the north pole, and let S = (0, 0, −1) be the south pole. We consider ϕ1 : S 2 − {N} → ℝ2 (i. e., U1 = S 2 − {N}) be the stereographic projection from the north pole given by ϕ1 (x, y, z) = (

y x , ). 1−z 1−z

2 2 Then ϕ−1 1 : ℝ → S is given by

ϕ−1 (u, v) = (

2u 2v u2 + v2 − 1 , , ). 1 + u2 + v2 1 + u2 + v2 1 + u2 + v2

Similarly, let ϕ2 : S 2 − {S} → ℝ2 (i. e., U2 = S 2 − {S}) be the stereographic projection from the south pole given by

1.4 The abstract surfaces

ϕ2 (x, y, z) = (

�

17

y x , ). 1+z 1+z

Then ϕ1 (U1 ∩ U2 ) = ℝ2 − {(0, 0)}, and (ϕ2 ∘ ϕ−1 1 )(u, v) = (

u2

v u , 2 ), 2 + v u + v2

which is a smooth map. In the same way, we can check that ϕ1 ∘ϕ−1 2 is also a smooth map. Hence S 2 is a two-dimensional smooth manifold. Smooth functions and smooth maps Let M be a smooth differential manifold of dimension m. A functions f : M → ℝ is said to be smooth at p ∈ M if there is a coordinate chart (U, ϕ) such that f ∘ ϕ−1 : ϕ(U) ⊂ ℝm → ℝ is smooth (i. e., its partials of any order exist). Similarly, let M and N be smooth differential manifolds of dimensions m and n. A map f : M → N is said to be smooth at p ∈ M if there are a coordinate chart (U, ϕ) of p and a coordinate chart (V , ψ) of q = f (p) such that ψ ∘ f ∘ ϕ−1 : ϕ(U) ⊂ ℝm → ψ(V ) ⊂ ℝn is smooth. Tangent space Tp M Let M be a smooth differential manifold of dimension m, and let p ∈ M. A smooth curve in M passing through p ∈ M is a smooth map α : (−ϵ, ϵ) → M such that α(0) = p. The key is to define its tangent vector α′ (0). Here is the definition: Denote by Cp∞ (M) the set of real-valued smooth functions defined around p ∈ M; then α′ (0) is the map α′ (0) : Cp∞ (M) → ℝ such that, for any f ∈ Cp∞ (M), α′ (0)(f ) =

d (f ∘ α)(t)|t=0 . dt

A tangent vector vp to M at p ∈ M is vp = α′ (0) for some smooth curve α : (−ϵ, ϵ) → M with α(0) = p. The tangent space Tp M is the collection of the tangent vectors vp to M at the point p. We now discuss the standard basis for the tangent space Tp M. Let (U, ϕ) be a local coordinate for M at p. Write ϕ = (x 1 , . . . , x m ) and the maps 𝜕x𝜕 i |p : Cp∞ (M) → ℝ for 1 ≤ i ≤ m as follows: 𝜕(f ∘ ϕ−1 ) 󵄨󵄨󵄨󵄨 𝜕 󵄨󵄨󵄨󵄨 (f ) := 󵄨 󵄨 , 𝜕x i 󵄨󵄨󵄨p 𝜕x i 󵄨󵄨󵄨ϕ(p)

f ∈ Cp∞ (M);

note that f ∘ ϕ−1 is a function defined around ϕ(p) ∈ ℝm , and (x 1 , . . . , x m ) is the standard −1

∘ϕ ) Cartesian coordinates of ℝm , so the partials 𝜕(f𝜕x |ϕ(p) are defined for 1 ≤ i ≤ m. Alteri natively, let (U, ϕ) be a local coordinate for M at p with ϕ(p) = 0, and let αi : (−ϵ, ϵ) → M, 1 ≤ i ≤ m, be the curves defined as αi (t) = (ϕ−1 ∘ γi )(t), where γ1 : t 󳨃→ (t, 0, . . . , 0), . . . , γm : t 󳨃→ (0, 0, . . . , t) are curves in ℝm . Then

18 � 1 Some background in differential geometry 𝜕 󵄨󵄨󵄨󵄨 ′ 󵄨 = αi (0), 𝜕x i 󵄨󵄨󵄨p and so

𝜕 | 𝜕x i p

∈ Tp M. It turns out that they indeed form a basis for Tp M, which is called

the standard basis for Tp M with respect to the coordinate system (U; x i ).

Metric and Gauss curvature for abstract surfaces A (Riemannian) metric on M is g which assigns, for every point p ∈ M, g(p) := ⟨⋅, ⋅⟩p , an inner product on Tp M such that g(p) is smooth as p varies. In terms of local coordinates (U, x i ), let gij = ⟨𝜕/𝜕x i , 𝜕/𝜕x j ⟩. Then we can write g = ∑ gij dx i dx j , where dx i dx j = 21 (dx i ⊗ dx j + dx j ⊗ dx i ) is a symmetric tensor. Let M be an abstract surface (two-dimensional manifold) with metric g. As above, we define the Christoffel symbols: in terms of local coordinates (U; x i ), Γkij =

𝜕gjt 𝜕gij 𝜕g 1 ∑ g kt ( itj + i − t ), 2 𝜕x 𝜕x 𝜕x

where (g kt ) is the inverse matrix of (gij ). Let Γm ij,k :=

𝜕Γm ij 𝜕x k

.

Write, for 1 ≤ i, j, k, m ≤ 2, the Riemann curvature tensor of the first kind 2

m m l m l m Rm ijk := Γik,j − Γjk,i + ∑(Γik Γjl − Γjk Γil ), l=1

and define the curvature tensor of the second kind 2

Rijkl := ∑ Rm ijk gml . m=1

Then the Gauss curvature is defined by K=

R1212 . det(gij )

(1.12)

S. S. Chern proved that the Gauss–Bonnet theorem still holds for a compact abstract surface without boundary.

2 Analysis and geometry of complex functions 2.1 Holomorphic functions Denote by ℂ the set of complex numbers and by z = x + iy complex variables, where i = √−1 and (x, y) ∈ ℝ2 . Let U ⊂ ℂ be a domain. Let f : U → ℂ be a complex-valued function on U. Let a ∈ U. The function f is said to be complex differentiable at a if the complex derivative f ′ (a) := lim

h→a

f (a + h) − f (a) h

exists. f is said to be holomorphic on D if f ′ (a) exists for every a ∈ D. Proposition 2.1.1 (The Cauchy–Riemann equations). Let f = u + iv. If f is holomorphic on D, then u and v satisfy the Cauchy–Riemann equations 𝜕u 𝜕v = , 𝜕x 𝜕y

𝜕u 𝜕v =− . 𝜕y 𝜕x

Conversely, suppose that u and v have continuous first-order partial derivatives with respect to x and y on D and u and v satisfy the Cauchy–Riemann equations on D, then f is holomorphic on D. Proof. “󳨐⇒” Since f ′ (z) exists, taking the limit as h → 0 with real h, we have f ′ (z) =

𝜕u 𝜕v +i . 𝜕x 𝜕x

Letting h = ik with real k and letting k → 0, we similarly obtain f (z + ik) − f (z) 𝜕u 𝜕v = −i + . k→0 ik 𝜕y 𝜕y

f ′ (z) = lim

Equating the two expressions for f ′ (z), we obtain our result. “⇐󳨐” Conversely, since the partials of u and v are continuous, by a multivariable calculus theorem, u and v are differentiable at (x, y). This means that u(x + h, y + k) − u(x, y) =

𝜕u 𝜕u h+ k + Ru (h, k), 𝜕x 𝜕y

and similarly for v, where Ru (h, k)/(h + ik) → 0 as h + ik → 0. Then f (z + h + ik) − f (z) = ( so https://doi.org/10.1515/9783110989557-002

𝜕u 𝜕v + i )(h + ik) + Ru (h, k) + iRv (h, k), 𝜕x 𝜕x

20 � 2 Analysis and geometry of complex functions f (z + h + ik) − f (z) 𝜕u 𝜕v = +i . h+ik→0 h + ik 𝜕x 𝜕x lim

Define the complex differential operators 𝜕 𝜕 1 𝜕 𝜕 = ( − √−1 ), = 𝜕z 2 𝜕x 𝜕y 𝜕z̄ √−1 d = 𝜕 + 𝜕,̄ d c = (𝜕̄ − 𝜕), 4π

1 𝜕 √ 𝜕 𝜕 ( + −1 ), 𝜕 = dz, 2 𝜕x 𝜕y 𝜕z √−1 so that dd c = 𝜕𝜕.̄ 2π

𝜕 𝜕̄ = d z,̄ 𝜕z̄

Corollary 2.1.2. Let f = u + iv. Assume that f is holomorphic on D. Then 𝜕f = 0. 𝜕z̄ Furthermore, we have f ′ (z) =

𝜕f . Conversely, if the partials of u and v are continuous and 𝜕z

𝜕f = 0, 𝜕z̄ then f is holomorphic on D. We now introduce the Cauchy theorem. Let γ : [a, b] → ℂ be a piecewise differentiable arc with γ(t) = z(t). We define b

∫ f (z)dz = ∫ f (z(t))z′ (t)dt. γ

a

Note that this integral is invariant under changes of parameter. We will use the following estimate several times: 󵄨󵄨 b 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 b 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 ′ 󵄨 ′ 󵄨󵄨∫ f (z)dz󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ f (γ(t))γ (t)dt 󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨 f (γ(t))󵄨󵄨󵄨󵄨󵄨󵄨γ (t)󵄨󵄨󵄨dt 󵄨󵄨 󵄨󵄨 γ 󵄨󵄨 󵄨󵄨󵄨 a 󵄨 a b

󵄨 󵄨 󵄨 󵄨 ≤ (sup󵄨󵄨󵄨 f (z)󵄨󵄨󵄨) ∫󵄨󵄨󵄨γ′ (t)󵄨󵄨󵄨dt. z∈γ

a

b

Observe that ∫a |γ′ (t)|dt is the arc length of γ. Let R be a closed rectangle with boundary 𝜕R. We say that a function is holomorphic on R if it is holomorphic on a region containing R. Theorem 2.1.3 (Cauchy’s theorem). If f is holomorphic on R, then ∫ f (z)dz = 0. 𝜕R

2.1 Holomorphic functions

� 21

Proof. For any rectangle S, let η(S) = ∫𝜕η f (z)dz. If we divide R into rectangles R1 , R2 , R3 , R4 , then η(R) = η(R1 )+⋅ ⋅ ⋅+η(R4 ), since the integrals over common sides cancel each other. For at least one rectangle Ri , we must have |η(Ri )| ≥ 41 |η(R)|. We denote this rectangle S1 and now divide S1 into four congruent rectangles and repeat the process. We thus arrive at a sequence S0 = R, S1 , S2 , . . . such that Si+1 ⊂ Si and |η(Si+1 )| ≥ 41 |η(Si )|, and hence |η(Sn )| ≥ 4−n |η(R)|. Let z∗ be the point of intersection of the Si for i = 1, 2, . . . . Given ϵ > 0, we can choose δ > 0 such that f (z) is analytic in |z − z∗ | < δ and 󵄨󵄨 f (z) − f (z∗ ) 󵄨 󵄨󵄨 ′ ∗ 󵄨󵄨󵄨 − f (z ) 󵄨󵄨 󵄨󵄨 < ϵ 󵄨󵄨 z − z∗ 󵄨󵄨 on |z − z∗ | < δ. If n is large enough, then Sn is contained in |z − z∗ | < δ. Now noticing that ∫S dz = 0 and ∫S zdz = 0, we have n

n

η(Sn ) = ∫ (f (z) − f (z∗ ) − f ′ (z∗ )(z − z∗ ))dz, 𝜕Sn

and it follows that 󵄨󵄨 󵄨 󵄨 ∗󵄨 󵄨󵄨η(Sn )󵄨󵄨󵄨 ≤ ϵ ∫ 󵄨󵄨󵄨z − z 󵄨󵄨󵄨|dz|. 𝜕Sn

In this last integral, |z − z∗ | ≤ c1 2−n , and | ∫𝜕S |dz||, the length of the perimeter of Sn , is n

≤ c2 2−n , so the last integral is bounded by c1 c2 4−n ϵ. This implies that |η(R)| ≤ c1 c2 ϵ; since ϵ is arbitrary, η(R) must be 0.

The following is the Cauchy theorem in a disk. It is sometimes referred to as the Cauchy–Goursat theorem. Theorem 2.1.4 (Cauchy–Goursat theorem). Let Dρ (z0 ) be the open disk |z − z0 | < ρ. If f is analytic in Dρ (z0 ), then ∫γ f (z)dz = 0 for every closed curve γ in Dρ (z0 ). Now we introduce the Cauchy integral formula. It is easy to verify that if a piecewise differentiable closed curve γ does not pass through the point a, then the value of the dz is a multiple of 2πi. We define the index of the point a with respect to γ or integral ∫γ z−a the winding number of γ with respect to a by n(γ, a) :=

1 dz . ∫ 2πi z − a γ

Theorem 2.1.5 (Cauchy’s integral formula). Let f be a holomorphic function in a domain D, and let γ be a closed curve in D. If a ∈ ̸ γ, then

22 � 2 Analysis and geometry of complex functions n(γ, a)f (a) =

f (z) 1 dz. ∫ 2πi z − a γ

Proof. Let us apply Cauchy’s theorem to F(z) = (f (z) − f (a))/(z − a). F is analytic for z ≠ a. For z = a, it satisfies the condition limz→a F(z)(z − a) = limz→a (f (z) − f (a)) = 0. Hence ∫ γ

f (z) − f (a) dz = 0, z−a

which is the same as ∫ γ

f (z) dz dz = f (a) ∫ . z−a z−a γ

Theorem 2.1.6 (Power series expansion). Let f : D → ℂ be holomorphic. Then for each z0 ∈ D, it can be represented as the power series ∞

f (z) = ∑ cn (z − z0 )n n=0

with nonzero radius of convergence R > 0, and cn =

1 2πi

∫ 𝜕BR (z0 )

f (w)dw , (w − z0 )n+1

where BR (z0 ) ⊂ D is the open disc centered at z0 with radius R. Moreover, for each specific point z0 ∈ D, the radius of convergence can be chosen as the minimum distance from point z0 to the boundary of D. Remark. For this reason, the holomorphic functions are also called analytic functions. Proof. Assume that f is holomorphic in some ball Br (z0 ). Let g(z) = f (z + z0 ). Then g is holomorphic in the ball Br (0). By Cauchy’s formula g(z) =

g(w)dw 1 . ∫ 2πi w−z 𝜕Br (0)

We have n

1 1 1 1 ∞ z = = ∑( ) , w − z w 1 − wz w n=0 w where the last series converges uniformly for all |z| < r since |z/w| < 1 by construction. Since the series converges uniformly and all the terms are continuous, we may interchange the order of summation and integration:

2.1 Holomorphic functions

∞

g(z) = ∑ zn ( n=0

� 23

g(w)dw 1 ). ∫ 2πi wn+1 𝜕Br (0)

Rreturn to the original function f , we conclude that ∞

f (z) = g(z − z0 ) = ∑ cn (z − z0 )n , n=0

where cn =

and

1 2πi

∫ 𝜕Br (z0 )

f (w)dw . (w − z0 )n+1

So we see that if f is holomorphic in D, then the derivatives of all orders for f exist,

f (z) =

f (ζ )dζ 1 , ∫ 2πi ζ −z 𝜕D

f (ζ )dζ 1 f (z) = , ∫ 2πi (ζ − z)2 ′

𝜕D

f

(n)

f (ζ )dζ n! . (z) = ∫ 2πi (ζ − z)n+1

(2.1)

𝜕D

Theorem 2.1.7 (Liouville’s theorem). If f is analytic and bounded in ℂ, then f is constant. Proof. If |f | is bounded by M, then, by (2.1) with D(r) being the disk of radius r, 󵄨󵄨 ′ 󵄨󵄨 −1 󵄨󵄨 f (z)󵄨󵄨 ≤ Mr . Letting r → ∞, we see that f ′ ≡ 0. Hence f is constant. Assume that f is holomorphic on an open subset of ℂ with a zero of order m at z0 , that is, f (z) = (z − z0 )m h(z) locally, and h(z0 ) ≠ 0. Fix some ρ > 0 such that h(z) ≠ 0 in 0 < |z − z0 | ≤ ρ. Then, on 0 < |z − z0 | ≤ ρ, f′ h′ = m(z − z0 )−1 + . f h Thus, by Cauchy’s theorem and Cauchy’s integral formula, m=

1 2πi

∫ |z−z0 |=ρ

f ′ (z) dz. f (z)

(2.2)

24 � 2 Analysis and geometry of complex functions Theorem 2.1.8 (Hurwitz’s theorem). Let {fk } be a sequence of holomorphic functions on a connected open set D ⊂ ℂ that converges uniformly on compact subsets of D to a holomorphic function f that is not constantly zero on D. If f has a zero of order m at z0 , then for every small enough ρ > 0 and for sufficiently large k ∈ ℕ (depending on ρ), fk has precisely m zeroes in the disk |z − z0 | ≤ ρ, including multiplicity. Furthermore, these zeroes converge to z0 as k → ∞. Proof. Fix some ρ > 0 such that f (z) ≠ 0 in 0 < |z − z0 | ≤ ρ. Choose δ > 0 such that |f (z)| > δ for z on the circle |z − z0 | = ρ. Since fk (z) converges uniformly on the disc we have chosen, we can find N such that |fk (z)| ≥ δ2 for every k ≥ N and every z on the circle, ensuring that the quotient fk′ (z)/fk (z) is well defined for all z on the circle |z − z0 | = ρ. By (2.1) we can prove that fk′ → f ′ uniformly on the disc |z − z0 | < ρ, and hence we have another uniform convergence: fk′ (z) f ′ (z) → . fk (z) f (z) Denote the number of zeros of fk (z) in the disk by Nk . By (2.2), m=

1 2πi

∫ |z−z0 |=ρ

f ′ (z) 1 dz = lim k→∞ 2πi f (z)

∫ |z−z0 |=ρ

fk′ (z) dz = lim Nk . k→∞ fk (z)

In the above step, we were able to interchange the integral and limit because of the uniform convergence of the integrand. We have shown that Nk → m as k → ∞. Since the Nk are integer valued, they must equal m for large enough k.

2.2 Harmonic functions A function u is harmonic in a region D ⊂ ℝ2 if it is of class C 2 (this means that u and its first- and second-order partial derivatives are continuous) and △u =

𝜕2 u 𝜕2 u + = 0. 𝜕x 2 𝜕y2

This equation is called the Laplace equation. If u is harmonic, then v is its conjugate harmonic function if u + iv is analytic. In general, there is no single-valued conjugate harmonic function, but there will be in a disk. To show this, let g(z) =

𝜕u 𝜕u −i , 𝜕x 𝜕y

which is analytic because the Cauchy–Riemann equations are satisfied. Therefore ∫γ g(z)dz = 0 for every closed curve γ in the disk. Now we can write

2.3 The Ahlfors–Schwarz lemma and the negative curvature method

g(z)dz = (

� 25

𝜕u 𝜕u 𝜕u 𝜕u dx + dy) + i(− dx + dy). 𝜕x 𝜕y 𝜕y 𝜕x

The real part is the differential of u, and the integral over every closed curve of du is 0. Therefore the integral over every closed curve of the imaginary part is 0. Define v(z) = ∫(− γ

𝜕u 𝜕u dx + dy), 𝜕y 𝜕x

where γ is any curve starting from 0, ending at z, and contained in the unit disk. So v is well defined and has the partials −𝜕/𝜕y and 𝜕u/𝜕x. It is easy to check that v satisfies the Laplace equation and that v is the conjugate harmonic function to u. Harmonic functions have the mean value property. Theorem 2.2.1 (Mean value theorem). If u is harmonic in a disk and |z − z0 | = r is contained in that disk, then 2π

1 u(z0 ) = ∫ u(z0 + reiθ )dθ. 2π 0

Proof. By a change of coordinates we may suppose z0 = 0. Let f be the analytic function which has u as its real part. By Cauchy’s integral formula we have 2π

f (z) f (reiθ ) iθ 1 1 f (0) = ire dθ. dz = ∫ ∫ 2πi z 2πi reiθ 0

|z|=r

The result follows by taking the real parts.

2.3 The Ahlfors–Schwarz lemma and the negative curvature method Theorem 2.3.1 (The maximum principle). If f is analytic and nonconstant in a region D ⊂ ℂ, then 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 f (z)󵄨󵄨󵄨 < sup󵄨󵄨󵄨 f (ζ )󵄨󵄨󵄨. ζ ∈D

Proof. If supζ ∈D |f (ζ )| = +∞, then the theorem obviously holds. Thus we assume that 󵄨 󵄨 sup󵄨󵄨󵄨 f (ζ )󵄨󵄨󵄨 = M < +∞. ζ ∈D

Assume there exists z0 ∈ D such that |f (z0 )| = M. If |f (z)| ≡ M, then it is easy to prove that f is also constant. Thus there is z1 ∈ D such that |f (z1 )| < M. Because D is connected, there is a continuous curve L : z = z(t), t ∈ [0, 1], inside D with z(0) = z0 and z(1) = z1 .

26 � 2 Analysis and geometry of complex functions Let 󵄨 󵄨 t ∗ = sup{t | 󵄨󵄨󵄨 f (z(t))󵄨󵄨󵄨 = M}. Since f and z(t) are both continuous, we know that |f (z(t ∗ ))| = M, t ∗ < 1, and for t ∈ [t ∗ , 1], 󵄨󵄨 󵄨 󵄨󵄨 f (z(t))󵄨󵄨󵄨 < M. Let z∗ = z(t ∗ ) ∈ D. Then there exists δ > 0 such that B(z∗ , δ) := {z | |z − z∗ | ≤ δ} ⊂ D. By the mean value formula (Cauchy’s integral formula), 2π

1 f (z ) = ∫ f (z∗ + reiθ )dθ. 2π ∗

0

Thus 2π

1 󵄨󵄨 ∗ 󵄨 󵄨 󵄨 M = 󵄨󵄨󵄨 f (z∗ )󵄨󵄨󵄨 ≤ ∫ 󵄨 f (z + reiθ )󵄨󵄨󵄨dθ, 2π 󵄨 0

that is, 2π

1 󵄨 󵄨 ∫ (󵄨󵄨 f (z∗ + reiθ )󵄨󵄨󵄨 − M)dθ ≥ 0. 2π 󵄨 0

Because |f (z∗ + reiθ )| − M is a nonpositive function, we get |f (z∗ + reiθ )| − M ≡ 0, that is, |f (z)| = M for all z ∈ B(z∗ , δ). Hence f (z) = Meiα for some real number α. On the other hand, since z(t) is continuous at t ∗ , z∗ = z(t ∗ ), there exists t1 ∈ (t ∗ , 1) such that z(t1 ) ∈ B(z∗ , δ), and thus, by the above, |f (z(t1 ))| < M, a contradiction. Hence 󵄨󵄨 󵄨 󵄨󵄨 f (z)󵄨󵄨󵄨 < M,

z ∈ D.

Theorem 2.3.2 (Schwarz lemma). Let D(1) be the unit disk on ℂ. Let f : D(1) → D(1) be a holomorphic map such that f (0) = 0. Then |f (z)| ≤ |z| for all z ∈ D(1). Proof. Define g(z) = f (z) if z ≠ 0 and g(0) = f ′ (0). Then g is holomorphic on D(1). Now z if D(r) = {z | |z| ≤ r} denotes the closed disk of radius r centered at the origin, then the maximum modulus principle implies that for r < 1, given any z ∈ D(r), there exists zr on the boundary of D(r) = {z | |z| ≤ r} such that 󵄨󵄨 󵄨 󵄨 󵄨 | f (zr )| 1 ≤ . 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨g(zr )󵄨󵄨󵄨 = |zr | r This proves the theorem by letting r → 1.

2.3 The Ahlfors–Schwarz lemma and the negative curvature method

� 27

Theorem 2.3.3 (Schwarz–Pick lemma). Let f : D(1) → D(1) be a holomorphic map. Then for any two points z1 , z2 ∈ D(1), we have 󵄨󵄨 f (z ) − f (z ) 󵄨󵄨 󵄨󵄨 z − z 󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 1 2 󵄨󵄨 󵄨 ≤ 󵄨󵄨 1 󵄨󵄨 󵄨. 󵄨󵄨 1 − f (z )f (z ) 󵄨󵄨󵄨 󵄨󵄨󵄨 1 − z̄1 z2 󵄨󵄨󵄨 1 2 In particular, for any z ∈ D(1), | f ′ (z)| 1 ≤ . 2 1 − | f (z)| 1 − |z|2 Proof. The proof of the Schwarz–Pick theorem follows from the Schwarz lemma and the fact that a Möbius transformation of the form z − z0 , z0̄ z − 1

|z0 | < 1,

maps the unit circle into itself. Fix z1 and define the Möbius transformations M(z) =

z1 − z , 1 − z1̄ z

ϕ(z) =

f (z1 ) − z

1 − f (z1 )z

.

Since M(z1 ) = 0 and the Möbius transformation is invertible, the composition ϕ ∘ f ∘ M −1 maps 0 to 0, and the unit disk is mapped into itself. Thus we can apply Theorem 2.3.2 (Schwarz’s lemma): −1 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 f (z1 ) − f (M (z)) 󵄨󵄨󵄨 −1 󵄨󵄨(ϕ ∘ f ∘ M )(z)󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨 ≤ |z|. 󵄨󵄨 1 − f (z )f (M −1 (z)) 󵄨󵄨󵄨 1

Now denoting z2 = M −1 (z) (which will still be in the unit disk) yields the desired conclusion 󵄨󵄨 f (z ) − f (z ) 󵄨󵄨 󵄨󵄨 z − z 󵄨󵄨 󵄨󵄨 󵄨 1 2 󵄨󵄨 2 󵄨󵄨 󵄨 ≤ 󵄨󵄨 1 󵄨. 󵄨󵄨 󵄨󵄨 1 − f (z )f (z ) 󵄨󵄨󵄨 󵄨󵄨󵄨 1 − z̄1 z2 󵄨󵄨󵄨 1 2 To reformulate the lemma in terms of geometric quantities, we introduce the following definition. Definition 2.3.4. Let Ω ⊂ ℂ be a region. A metric on Ω is ds2 = 2a(z)dzd z̄ with smooth a > 0; ds2 = 2b(z)dzd z̄ is said to be a pseudo-metric on Ω if for every z0 ∈ Ω, b(z) = (z−z0 )k h(z) around z0 , where k ≥ 0, and h > 0 is smooth near z0 . The Gaussian curvature function Kds2 : Ω → [−∞, ∞) of ds2 is defined by Kds2 (z) = −

1 𝜕2 log b(z) b(z) 𝜕z𝜕z̄

for z ∈ U-Zero(b). For z ∈ Zero(b), we define Kds2 (z) = −∞.

28 � 2 Analysis and geometry of complex functions Example. Let D(r) be the disc of radius r on ℂ with center at the origin. The metric ds2 =

4r 2 dzd z̄ (r 2 − |z|2 )2

(2.3)

is called the Poincaré metric on D(r). We have K ≡ −1. We can reformulate the Schwarz–Pick lemma in terms of the Poincaré metric on the unit disc D(1). Lemma 2.3.5 (Schwarz–Pick). Let gP be the Poincaré metric on the unit disc D(1), and let f : D(1) → D(1) be a holomorphic map. Then f ∗ gP ≤ gP , that is, the Poincaré distance decreases under any holomorphic map. We now prove the more general Ahlfors–Schwarz lemma, which extends the Schwarz–Pick lemma. Theorem 2.3.6 (Ahlfors–Schwarz lemma). Let ds2 denote the Poincaré metric on the unit disc D(1). Let dσ 2 be a pseudo-metric on D(1) with Gaussian curvature bounded above by −1. Then dσ 2 ≤ ds2 . Proof. Let D(r) be the disc of radius r < 1 with the Poincaré metric ds2 of curvature −1 given by ds2 = 2ar (z)dzd z,̄

where ar (z) =

2r 2 . (r 2 − |z|2 )2

We compare this metric with dσ 2 = 2b(z)dzd z.̄ Put μ(z) = log

b(z) . ar (z)

Since μ(z) → −∞ as z → 𝜕D(r), there is a point z0 ∈ D(r) such that μ(z0 ) = sup{μ(z); z ∈ D(r)} > −∞. Then b(z0 ) > 0. Since z0 is a maximal point of μ(z), 0≥

𝜕2 μ (z ). 𝜕z𝜕z̄ 0

On the other hand, since the Gaussian curvature of the Poincaré metric is −1 and the curvature of dσ 2 is bounded above by −1,

2.3 The Ahlfors–Schwarz lemma and the negative curvature method

𝜕2 log ar = ar (z) and 𝜕z𝜕z̄

� 29

𝜕2 log b (z) ≥ b(z). 𝜕z𝜕z̄

So 0≥

𝜕2 log ar 𝜕2 μ 𝜕2 log b (z0 ) = (z0 ) − (z0 ) ≥ b(z0 ) − ar (z0 ). 𝜕z𝜕z̄ 𝜕z𝜕z̄ 𝜕z𝜕z̄

Hence ar (z0 ) ≥ b(z0 ), and so μ(z0 ) ≤ 0. By the choice of z0 we have μ(z) ≤ 0 on D(r), that is, ar (z) ≥ b(z). The theorem is proved by letting r → 1. To prove the little Picard theorem, we introduce the chordal distance on ℂ ∪ {∞} by |w1 , w2 | =

|w1 − w2 | √1 + |w1 |2 √1 + |w2 |2

if w1 , w2 ≠ ∞ and |w1 , w2 | =

1 √1 + |w1 |2

if w1 ≠ ∞ and w2 = ∞. Lemma 2.3.7. Let g be a meromorphic function on D(R) = {|z| < R} (0 < R ≤ +∞) (i. e., g is holomorphic on D(R) except some poles). Let α ∈ ℂ. Then, for a fixed ϵ > 0, there exists δ0 > 0 such that for all δ ≥ δ0 , on D(R)\{z | g(z) = α}, |g ′ |2 𝜕2 1 1 log ≥ ( − ϵ) 𝜕z𝜕z̄ log(δ/|g, α|2 ) (1 + |g|2 )2 |g, α|2 log2 (δ/|g, α|2 ) Proof. Let ϕ := |g, α|2 =

|g − α|2 . (1 + |g|2 )(1 + |α|2 )

Then we have ̄ + αg)̄ 𝜕ϕ g ′ (ḡ − α)(1 = , 2 2 𝜕z (1 + |g| ) (1 + |α|2 ) and so 󵄨󵄨 𝜕ϕ 󵄨󵄨2 |g ′ |2 |g − α|2 󵄨󵄨 󵄨󵄨 ̄ 2. |1 + αg| 󵄨󵄨 󵄨󵄨 = 󵄨󵄨 𝜕z 󵄨󵄨 (1 + |g|2 )4 (1 + |α|2 )2

30 � 2 Analysis and geometry of complex functions Since 1−ϕ=

̄ 2 |1 + αg| , (1 + |g|2 )(1 + |α|2 )

we obtain 󵄨󵄨 𝜕ϕ 󵄨󵄨2 |g ′ |2 󵄨󵄨 󵄨󵄨 2 . 󵄨󵄨 󵄨󵄨 = (ϕ − ϕ ) 󵄨󵄨 𝜕z 󵄨󵄨 (1 + |g|2 )2 On the other hand, |g ′ |2 𝜕2 log(1 + |g|2 ) = . 𝜕z𝜕z̄ (1 + |g|2 )2 Thus 𝜕2 1 𝜕 1 𝜕 log = ( log ϕ) 𝜕z𝜕z̄ log(δ/ϕ) 𝜕z log(δ/ϕ) 𝜕z̄ = = =

󵄨󵄨 𝜕ϕ 󵄨󵄨2 𝜕2 1 −1 󵄨󵄨 󵄨󵄨 log(1 + |g|2 ) + 󵄨󵄨 󵄨󵄨 log(δ/ϕ) 𝜕z𝜕z̄ ϕ2 log2 (δ/ϕ) 󵄨󵄨 𝜕z 󵄨󵄨 ϕ − ϕ2 |g ′ |2 1 ( − ) 2 2 (1 + |g| ) ϕ2 log2 (δ/ϕ) log(δ/ϕ)

|g ′ |2 1 1 1 )). ( −( 2 + 2 2 2 (1 + |g| ) ϕ log (δ/ϕ) log (δ/ϕ) log(δ/ϕ)

We can take δ = δ0 such that log−2 (δ/ϕ) + log−1 (δ/ϕ) < ϵ because ϕ ≤ 1. Thus we get the desired inequality. Theorem 2.3.8. Let g be a nonzero meromorphic function on D(R) = {|z| < R}. Assume that its image omits α1 , . . . , αq ∈ ℂ ∪ {∞}. If q > 2, then there exist constants δ > 0 and C > 0 such that q

|g ′ | 1 R ≤C 2 . ∏ (1 + |g|2 ) j=1 |g, αj | log(δ/|g, αj |2 ) R − |z|2 Proof. After a change of g by a suitable Möbius transformation, we may assume that aq = ∞. Let v := Then

q

|g ′ | 1 . ∏ 2 (1 + |g| ) j=1 |g, αj | log(δ/|g, αj |2 )

2.3 The Ahlfors–Schwarz lemma and the negative curvature method

v2 =

(1 + |g|2 )q−1 w

q ∏j=1 log2 (δ/|g, αj |2 )

� 31

,

where q−1

w=

|g ′ |2 ∏j=1 (1 + |αj |2 ) q−1

∏j=1 |g − αj |2

.

Then for ϵ = (q − 1)/q, take δ such that the above lemma holds for all αj . Thus, also using |g ′ |2 𝜕2 log(1 + |g|2 ) = 𝜕z𝜕z̄ (1 + |g|2 )2 and noticing that log w is harmonic, we have q

|g ′ |2 q−1 𝜕2 2 log v2 ≥ (q − 1 + ∑( − )) 2 2 2 2 2 ̄ 𝜕z𝜕z q (1 + |g| ) j=1 |g, αj | log (δ/|g, αj | ) q

=

2|g ′ |2 1 . ∑ (1 + |g|2 )2 j=1 |g, αj |2 log2 (δ/|g, αj |2 )

For every fixed z ∈ DR , it is easy to see that there is at most one j (denoted by j0 ) such that |g(z), αj | ≥ mink 0 such that |g ′ |2 𝜕2 1 log v ≥ 𝜕z𝜕z̄ (1 + |g|2 )2 |g, αj |2 log2 (δ/|g, αj |2 ) 0 ≥C

q

|g ′ |2 1 = Cv2 . ∏ (1 + |g|2 )2 j=1 |g, αj |2 log2 (δ/|g, αj |2 )

The theorem thus follows from the Ahlfors–Schwarz lemma. Corollary 2.3.9. Let g be a nonzero meromorphic function on D(R) = {|z| < R}. Assume that its image omits α1 , . . . , αq ∈ ℂ∪{∞} with aq = ∞. If q > 2, then there exists a constant C > 0 such that for 0 < η < 1, |g ′ |(1 + |g|2 ) q−1

q−2−qη 2

∏j=1 |g − αj |1−η

≤C

R . R2 − |z|2

(2.4)

Proof. Because the function log(δx 2 )/x η (1 ≤ x < +∞) is bounded, we can replace log2 (δ/|g, αj |2 ) by |g, αj |2η and thus obtain that |g ′ | 1 R ≤C 2 . (1 + |g|2 ) ∏qj=1 |g, αj |1−η R − |z|2

32 � 2 Analysis and geometry of complex functions Using the fact aq = ∞ and using the definition of the chordal distance that |w1 , w2 | = |w1 −w2 |

√1+|w1 |2 √1+|w2 |2

if w1 , w2 ≠ ∞ and |w1 , w2 | =

identity

1

√1+|w1 |2

if w1 ≠ ∞ and w2 = ∞, we have the

q−2−qη

′ 2 |g ′ | 1 ′ |g |(1 + |g| ) 2 = C q q−1 (1 + |g|2 ) ∏j=1 |g, αj |1−η ∏ |g − αj |1−η j=1

q−1

1

with C ′ = ∏j=1 (1 + |αj |2 ) 2 (1−η) . Theorem 2.3.10 (Little Picard theorem). Let f be a meromorphic function on ℂ. If the image of f omits at least three points in ℂ ∪ {∞}, then f is constant. Proof. From (2.4), letting R → ∞, we obtain that f ′ ≡ 0. Thus f is constant. In the rest of the section, we introduce the results that will be used in the Chapter 5. The results can also be viewed as the dual form of the Ahlfors–Schwarz–Pick lemma. Also see [64, 65]. Lemma 2.3.11. Let D(r) be the disk of radius r, 0 < r < 1, and let R be the hyperbolic radius of D(r) in the unit disc (with respect to the Poincaré metric). Let ds2 = λ(z)2 |dz|2 be any conformal metric on D(r) such that the geodesic distance from z = 0 to |z| = r is greater than or equal to R. If the Gauss curvature K of the metric ds2 satisfies −1 ≤ K ≤ 0, then the distance of any point to the origin in the metric ds2 is greater than or equal to the hyperbolic distance. Note that the hyperbolic metric in the unit disk is given by ̂ 2 |dz|2 , d s2̂ = λ(z)

̂ = λ(z)

2 , 1 − |z|2

and has the curvature K̂ ≡ −1. The relation between the quantities R and r is therefore given by r

r

̂ R = ∫ λ(z)|dz| =∫ 0

0

1+r 2 dt = log , 1−r 1 − t2

and from Lemma 2.3.11 it follows that ̂ = log ρ(z) ≥ ρ(z)

1 + |z| , 1 − |z|

where ρ and ρ̂ represent the distances from the point z to the origin in the metric ds2 and the hyperbolic metric, respectively.

2.3 The Ahlfors–Schwarz lemma and the negative curvature method

� 33

Proof of Lemma 2.3.11. Note first that in the relation above between R and r, we have dR 2 = > 0, dr 1 − r 2 and we may solve for r in terms of R: r=

eR − 1 , eR + 1

(2.5)

̂ eρ(z) −1 , ̂ ρ(z) e +1

(2.6)

or, in general, |z| =

̂ where the right-hand side is increasing in ρ(z). We may apply the comparison theorem of Greene and Wu ([38], Prop. 2.15, p. 26) to the metric ds2 and the hyperbolic metric d s2̂ on the disk |z| ≤ r. The comparison theorem states that for any smooth monotone increasing function f , we have ̂ ∘ ρ), ̂ △(f ∘ ρ) ≤ △(f where ρ and ρ̂ are the distances to the origin in the metrics ds2 and d s2̂ , respectively, △ and △̂ are the Laplacians with respect to the two metrics, and the two sides are evaluated at points of the same level sets of the two metrics, that is, ρ = c on the left, and ρ̂ = c on the right, provided in two dimensions that the Gauss curvatures K and K̂ satisfy 0 ≥ K ≥ K,̂ with a similar condition on the Ricci curvature in higher dimension. In our case, we have 0 ≥ K ≥ −1 = K,̂ and so we may apply the theorem. Note that the function log |z| = log

̂ eρ(z) −1 ̂ ρ(z) e +1

is harmonic with respect to z and is therefore also harmonic with respect to any conformal metric on 0 < |z| < 1. In other words, setting f (t) = log

et − 1 , et + 1

we have ̂ ∘ ρ)̂ ≡ 0 △(f for 0 < |z| < 1. Since f is increasing, we may apply the Greene–Wu comparison theorem to conclude that △(f ∘ ρ) ≤ 0

34 � 2 Analysis and geometry of complex functions for 0 < |z| < r, that is, f ∘ ρ is superharmonic. For z near 0, we have ρ(z) ∼ λ(0)|z|, and we may apply the minimum principle to the function log

1 1 eρ(z) − 1 f (ρ(z)) = log , |z| |z| eρ(z) + 1

which is superharmonic in 0 < |z| < r and bounded near the origin, to conclude that it takes on its minimum on the boundary |z| = r. Since ρ(z) ≥ R on |z| = r, by (2.5) we have that log

1 eρ(z) − 1 1 eR − 1 ≥ log R =0 ρ(z) |z| e re +1 +1

for |z| < r. Hence ̂ eρ(z) − 1 eρ(z) −1 ≥ |z| = ̂ eρ(z) + 1 eρ(z) +1

̂ proving the lemma. by (2.6), which implies that ρ(z) ≥ ρ(z), Lemma 2.3.11 also implies another dual form of the Ahlfors–Schwarz–Pick lemma, closer to the original one. Lemma 2.3.12. Let S be a simply connected surface with a complete metric ds2 whose Gauss curvature satisfies −1 ≤ K ≤ 0. If S is mapped conformally onto the unit disc, then the distance between any two points of S is greater than or equal to the hyperbolic distance between the corresponding points in the disk. Proof. Given two points p and q of S, we may map p onto the origin, and let z be the image of the point q. Then the distance between p and q on S is given by ρ(z) in terms of ̂ be the pullback of the metric on S onto the disk. For any r such that |z| < r < 1, let ρ(z) the hyperbolic distance from 0 to z, and let ρr (w) be the pullback of the metric on S to |w| < r under the map z = w/r. Then since S is complete, we may apply Lemma 2.3.11 to conclude that ̂ ≤ ρr (w) = ρr (rz). ρ(z) Since ρr (rz) → ρ(z) as r → 1, the lemma is proved. Note that Lemma 2.3.12 combined with the standard Ahlfors–Schwarz–Pick lemma implies a generalization of Ahlfors’ lemma due to Yau [91] (see also Troyanov [85]): Let S1 be a simply connected Riemann surface with a complete metric ds2 whose Gauss curvature satisfies −1 ≤ K ≤ 0, and let S2 be a Riemann surface with Gauss curvature bounded above by −1. Let f : S1 → S2 be a holomorphic map. Then f is distance decreasing.

2.4 The Riemann surfaces

� 35

2.4 The Riemann surfaces A Riemann surface M is a one-dimensional complex manifold, that is, it is covered by a family of coordinates {Uα , ϕα } that covers the totality of M, where ⋃α Uα = M, and ϕα : Uα → ℂ are functions with ϕα ∘ ϕ−1 β holomorphic on ϕβ (Uα ∩ Uβ ) (assuming that Uα ∩ Uβ ≠ 0). Let M be a Riemann surface with a local coordinate z = x + √−1y. Denote 𝜕 𝜕 1 𝜕 𝜕 = ( − √−1 ), = 𝜕z 2 𝜕x 𝜕y 𝜕z̄ √−1 (𝜕̄ − 𝜕) d = 𝜕 + 𝜕,̄ d c = 4π

1 𝜕 √ 𝜕 𝜕 ( + −1 ), 𝜕 = dz, 2 𝜕x 𝜕y 𝜕z √ −1 so that dd c = 𝜕𝜕.̄ 2π

𝜕 𝜕̄ = d z,̄ 𝜕z̄

The conformal metric on M is given by ds2 = 2a(dx 2 + dy2 ) = 2adzd z̄ locally, where a is a positive smooth function. Its Gaussian curvature is, according to (1.5), K =− where △ =

𝜕2 𝜕x 2

+

𝜕2 𝜕y2

1 △ log a, 4a

(2.7)

2

𝜕 = 4 𝜕z𝜕 is the usual Laplacian. Let z̄

ω = a(z)

√−1 dz ∧ d z̄ 2π

be its associated Kähler form (or metric form). With ω we associate the Ricci form Ric(ω) := dd c log a. Then we have Ric(ω) = dd c log a = −Kω.

(2.8)

Assume that M is compact (and without boundary). By the Gauss–Bonnet theorem in Chapter 1 we have − ∫ Ric(ω) = ∫ Kω = χ(M) = 2 − 2g, M

(2.9)

M

where g is the genus of M. In the study of holomorphic map f : M → N (for example, in deriving the Riemann–Hurwitz formula below), where M and N are compact Riemann surfaces, we take a smooth positive (1, 1)-form ω on N and pull it back to M through f to √ get a pseudo-positive (1, 1)-form ϕ = f ∗ ω, that is, we can write it locally as ϕ = h 2π−1 dz∧d z̄ 2ν on M with h(z) = |z| h0 (z) with smooth h0 > 0. In this case, we say that ϕ has a zero of order ν at z = 0. We have the following Gauss–Bonnet theorem for pseudo-metric, extending (2.9), by slightly modifying the proof (see [76, Theorem A1.1.1]). Theorem 2.4.1 (Gauss–Bonnet for pseudo-metric). Let M be a compact Riemann surface. √ Let ϕ = h 2π−1 dz ∧ d z̄ be a pseudo-positive (1, 1)-form on M. Let a1 , . . . , ak be the zeros of ϕ on M with vanishing order νi , 1 ≤ i ≤ k (see above for the definition). Then

36 � 2 Analysis and geometry of complex functions k

− ∫ Ric(ϕ) = 2 − 2g + ∑ νj . j=1

M

If we rewrite (in the sense of currents) k

∫ Ric[ϕ] = ∫ dd c [log h] := ∫ dd c log h + ∑ νj , M

M

j=1

M

then the theorem becomes ∫ Ric[ϕ] = 2g − 2.

(2.10)

M

We now turn to holomorphic maps between two Riemann surfaces. Let f : M → N be a holomorphic map, where M and N are compact Riemann surfaces. We call υf (p) the multiplicity of f at p ∈ M if there are local coordinates z for M at p ∈ M with z(p) = 0 and w for N at f (p), respectively, with w(f (p)) = 0 such that w = zυf (p) . Theorem 2.4.2 (Riemann–Hurwitz). Let M and N be compact Riemann surfaces, and let f : M → N be a holomorphic map. Then (2gM − 2) = deg(f )(2gN − 2) + ∑ (υf (p) − 1), p∈M

where gM and gN are the genera of M and N, respectively. Note that here deg(f ) is the number of f −1 {a} for some (hence for all) a ∈ N, counting the multiplicities. In other words, if we denote by |f −1 {a}| the cardinality of the set f −1 {a}, then 󵄨 󵄨 deg(f ) = 󵄨󵄨󵄨 f −1 {a}󵄨󵄨󵄨 +

∑ (υf (p) − 1).

p∈f −1 {a}

(2.11)

In the case N = ℙ1 (ℂ), let ωFS be the Fubini–Study form on ℙ1 (ℂ). Then ωFS = √−1 1 dw ∧ d w̄ = dd c log(1 + |w|2 ) for the affine coordinate (w, 1) ∈ ℙ1 (ℂ). Write (1+|w|2 )2 2π

f = f0 /f1 . Then f ∗ ωFS = dd c log(1 + |f0 /f1 |2 ). Let pl , 1 ≤ l ≤ k, be the (distinct) zeros of f1 on M, and choose holomorphic local coordinates zl with zl (pl ) = 0. We consider a sufficiently small positive ϵ such that U l (ϵ) := {zl | |zl | ≤ ϵ} are mutually disjoint. Since dd c log |f1 |2 = 0 outside {p1 , . . . , pk }, we obtain ∫ f ∗ ωFS = lim M

ϵ→0

= lim

ϵ→0

∫

f ∗ ωFS

M−⋃kl=1 Ul (ϵ) k

∫ M−⋃kl=1 Ul (ϵ)

dd c log(1 + | f0 /f1 |2 ) = − lim ∑ ∫ d c log(1 + | f0 /f1 |2 ) ϵ→0

l=1 𝜕U (ϵ) l

2.4 The Riemann surfaces k

� 37

k

= − lim ∑ ∫ d c log(| f0 |2 + | f1 |2 ) + lim ∑ ∫ d c log | f1 |2 ϵ→0

ϵ→0

l=1 𝜕U (ϵ) l

l=1 𝜕U (ϵ) l

k

= lim ∑ ∫ d c log | f1 |2 . ϵ→0

l=1 𝜕U (ϵ) l

On the other hand, k

k

∑ ∫ d c log | f1 |2 = ∑ l=1 𝜕U (ϵ)

l=1

l

k f ′ (z) 1 ∫ 1 dz = ∑ νf1 (pl ) = deg(f ). 2πi f1 (z) l=1 |zl |=ϵ

Hence deg(f ) = ∫ f ∗ ωFS .

(2.12)

M

Proof of Theorem 2.4.2. We only prove the case where N = ℙ1 (ℂ). Let ωFS be the Fubini– √ Study form on ℙ1 (ℂ). Then f ∗ ωFS = h 2π−1 dz ∧ d z̄ is a pseudo-positive (1,1)-form on M whose singular divisor is the ramification divisor (denoted by ram(f )). By (2.10), ∫ dd c [log h] = 2gM − 2, M

that is, ∫ dd c log h + deg(ram(f )) = 2gM − 2. M

Since by definition ωFS =

√−1 1 dw ∧ d w̄ = dd c log(1 + |w|2 ) 2 2 (1 + |w| ) 2π

for the affine coordinate (w, 1) ∈ ℙ1 (ℂ), we have Ric(ωFS ) = −2ωFS . Hence −2 deg(f ) = −2 ∫ f ∗ ωFS = ∫ f ∗ Ric(ωFS ) = ∫ Ric(f ∗ ωFS ) M

M

M

= ∫ dd c log h = 2gM − 2 − deg(ram(f )) = 2gM − 2 − ∑ (υ(p) − 1). p∈M

M

For any subset E ⊂ M, define r(E) := ∑ (υf (p) − 1). p∈E

38 � 2 Analysis and geometry of complex functions Let a1 , . . . , aq ∈ N, and let E = f −1 {a1 , . . . , aq }. Then from (2.11) we have q deg(f ) = |E| + r(E), where |E| is the cardinality of the set E. By applying the Riemann–Hurwitz theorem we have (2gM − 2) = deg(f )(2gN − 2) + q deg(f ) − |E| + r(E c ). So we have derived the following theorem. Theorem 2.4.3 (Algebraic second main theorem (the Riemann relation)). Let M and N be compact Riemann surfaces, and let f : M → N be a nonconstant holomorphic map. Let a1 , . . . , aq be distinct points in N, and let E = f −1 {a1 , . . . , aq } ⊆ M. Then (q + 2gN − 2) deg(f ) ≤ |E| + 2(gM − 1). In particular, when N = ℙ1 (ℂ), we have (q − 2) deg(f ) ≤ |E| + 2(gM − 1).

2.5 The theory of algebraic curves In this section, we present the theory of algebraic curves. We mainly follow the method from Chapter 2 of the book Principle of algebraic geometry by Griffiths-Harris (see [39]). The n-dimensional complex projective space is ℙn (ℂ) = ℂn+1 − {0}/∼, where (a0 , a1 , . . . , an ) ∼ (b0 , b1 , . . . , bn ) if and only if (a0 , . . . , an ) = λ(b0 , . . . , bn ) for some λ ∈ ℂ. We denote by [a0 : ⋅ ⋅ ⋅ : an ] the equivalence class of (a0 , . . . , an ). By an algebraic curve we mean a holomorphic map f : M → ℙn (ℂ), where M is a compact Riemann surface. In this section, we study algebraic curves. The theory is regarded as a model for the general theory of holomorphic curves (Nevanlinna theory). Let M be a Riemann surface, and let f : M → ℙn (ℂ) be a holomorphic map. For every point p ∈ M, clearly, we can lift f locally to ℂn+1 , that is, in a neighborhood of any point p ∈ M, we can find a holomorphic vector-valued function f = (f0 , . . . , fn ) such that f = [f0 : ⋅ ⋅ ⋅ : fn ]. Such f is called a reduced representation of f at p if f0 , . . . , fn have no common zeros. Let f = (f0 , . . . , fn ) be a reduced representation of f . Define k+1

Fk = f ∧ f′ ∧ ⋅ ⋅ ⋅ ∧ f(k) : M → ⋀ ℂn+1 . Evidently, Fn+1 ≡ 0. The map Fk = ℙ(Fk ) : M → ℙ(⋀k+1 ℂn+1 ) ⊂ ℙNk , where Nk = (n+1)! − 1 and ℙ is the natural projection, is called the kth associated map of f . Note (k+1)!(n−k)! that Fk is well defined, that is, it is independent of the choice of lifting f and of the local coordinate z.

2.5 The theory of algebraic curves

� 39

Let {e0 = (1, . . . , 0), . . . , en = (0, . . . , 1)} be the standard basis of ℂn+1 . Then we can write Fk = =

∑

det(fj(l) )0≤t≤k,0≤l≤k ej0 ∧ ⋅ ⋅ ⋅ ∧ ejk

∑

W (fj0 , fj1 , . . . , fjk )ej0 ∧ ⋅ ⋅ ⋅ ∧ ejk ,

0≤j0 0, 1 ≤ λ ≤ n} and use (2.18), then λ

n−1

n

k=0

t=1

∑ (n − k)νk+1 (P) ≥ ∑(υlP,j (P) − t) t

≥ ∑ max{0, υlP,j (P) − n} λ

lj ∈IP λ

= ∑ (υlP,j (P) − min{n, υlP,j (P)}) λ

lj ∈IP

λ

λ

n

= ∑(υlP,j (P) − min{n, υlP,j (P)}) t=1 q

t

t

= ∑(υlj (P) − min{n, υlj (P)}). j=1

Combining this inequality with (2.19), we get n−1

n−1

n−1

k=0

k=0 P∈S

k=0 P∈S̸

∑ (n − k)βk+1 = ∑ ∑ (n − k)νk+1 (P) + ∑ ∑ (n − k)νk+1 (P)

(2.19)

46 � 2 Analysis and geometry of complex functions q

q

≥ ∑ ∑ υlj (P) − ∑ ∑ min{n, υlj (P)} − j=1 P∈M

q

j=1 P∈S̸

= q deg(f ) − ∑ ∑ min{n, υlj (P)} − j=1 P∈S̸

n(n + 1) |S| 2

n(n + 1) |S|, 2

where we used the fact that ∑p∈M υlj (P) = deg(f ) for each j. Applying this to the Brill– Segre formula proves the theorem. The degenerate case can be done by using the Nochka weights (see Section 4.3). We omit the details.

3 Minimal surfaces in ℝ3 In this chapter, we present the theory of value distirbutions of the Gauss maps of minimal surfaces in ℝ3 . For the references, see [63, 34, 3, 16].

3.1 The minimal surfaces A surface in ℝ3 is called a minimal surface if its mean curvature is identically zero. Recall that the mean curvature H = 21 (κ1 + κ2 ), where κ1 and κ2 are the principal curvatures. So, geometrically, a minimal surface has the following character: at each point, the two principal curvatures (the largest and smallest normal curvatures) are canceled out. Also note that the Gauss curvature K of a minimal surface is always nonpositive, since K = κ1 κ2 = −κ12 ≤ 0 when H = 21 (κ1 + κ2 ) = 0. Furthermore, if K ≡ 0, then κ1 = κ2 = 0, hence all normal curvatures are zero, and therefore the surface lies in a plane. Theorem 3.1.1. For a minimal surface M immersed in ℝ3 , its Gauss curvature K ≡ 0 if and only if it lies in a plane. –

The followings are some standard examples of minimal surfaces. The catenoid is given by x(u, v) = (a cosh v cos u, a cosh v sin u, av),

–

0 < u < 2π, −∞ < v < +∞.

This is a surface of revolution generated by rotating the catenary y = a cosh( az ) about the z-axis. It is easily checked that its mean curvature H ≡ 0. Thus the catenoid is a minimal surface. It can be characterized as the only surface of revolution that is minimal, that is, if a surface of revolution M is a minimal surface, then M is contained in either a plane or a catenoid. A helicoid is a ruled surface swept out by a straight line that rotates at constant speed about an axis perpendicular to the line while simultaneously moving at constant speed along the axis. We can take the axis to be the z-axis. Let ω be the angular velocity of the rotating line, and let α be its speed along the z-axis. If the line starts along the x-axis, at time v the center of the line is at (0, 0, αv), and it has rotated by the angle ωv. Hence the point of the line initially at (u, 0, 0) is now at the point with position vector x(u, v) = (u cos ωv, u sin ωv, αv). This is a parameterization of the helicoid. It is easily checked that its mean curvature H ≡ 0. Thus the helicoid is a minimal surface. It can be characterized as the only minimal surface, other than the plane, which is also a ruled surface; that is, any ruled minimal surface in ℝ3 is part of a plane or a helicoid.

https://doi.org/10.1515/9783110989557-003

48 � 3 Minimal surfaces in ℝ3 –

Enneper’s surface is defined by 1 1 x(u, v) = (u − u3 + uv2 , v − v3 + vu2 , u2 − v2 ). 3 3

–

It is easily checked that its mean curvature H ≡ 0. Thus Enneper’s surface is a minimal surface. Scherk’s surface is defined by the Cartesian equation z = ln(

–

cos y ). cos x

The graph z = f (x, y). In this case the parameterization is x(x, y) = (x, y, f (x, y)). Hence the coefficients of the first fundamental form are g11 = 1 + fx2 , g12 = fx fy , g22 = 1 + fy2 , and the coefficients of the second fundamental form are h11 =

fxx

√1 + fx2 + fy2

,

h12 =

fxy √1 + fx2 + fy2

,

h22 =

fyy √1 + fx2 + fy2

.

Therefore H=

fxx (1 + fy2 ) − 2fx fy fxy + (1 + fx2 )fyy 2(1 + fx2 + fy2 )3/2

.

The graph z = f (x, y) is a minimal surface if and only if H = 0, that is, (1 + fy2 )fxx − 2fx fy fxy + (1 + fx2 )fyy = 0. We will now try to understand why the word “minimal” is used for such surfaces. Consider the problem first posed in the mid-1800s by the Belgian physicist Plateau: given a curve C, find a minimal surface M having C as its boundary. As we will see, least-area surfaces are minimal. Thus another version of Plateau’s problem is to find a least-area surface having C as its boundary. What is a necessary condition that M have least-area among all surfaces with boundary C? The answer may be found in a simplified version of the calculus of variations as follows: Let M be the surface with parameterization x : U → ℝ3 . Let γ be a curve in U that bounds a subdomain △ ⊂ U. Consider the nearby surfaces that look like slightly deformed versions of M, Mλ : g(u, v) = x(u, v) + λh(u, v)n(u, v),

(u, v) ∈ △,

where λ is a real number, n(u, v) is the unit normal to the surface M, and h is a C 2 function in U that has the effect, when multiplied by a small λ and added to x, of moving points of M a small bit and leaving the curve γ fixed, that is, h vanishes on the curve γ. The

3.1 The minimal surfaces

� 49

surface Mλ is called a normal variation of M since we are varying the surface M via the parameter λ along the direction of our unit normal n. Denoting by Aλ the area of the surface Mλ , we will prove the following: Theorem 3.1.2.

dA 󵄨󵄨󵄨󵄨 2 1/2 = −2 ∬ hH(g11 g22 − g12 ) hdudv, 󵄨 dλ 󵄨󵄨󵄨λ=0 △

where H is the mean curvature of M, and △ is the interior of γ. Proof. Differentiating g with respect to u and v, we have gu = xu + λ(hnu + hu n),

gv = xv + λ(hnv + hv n).

λ λ λ If we let g11 , g12 , g22 be the first fundamental form of the surface Mλ , then from the above we get λ g11 = g11 − 2λhh11 + O(λ2 ),

F λ = F − 2λhh12 + O(λ2 ),

and λ g22 = g22 − 2λhh22 + O(λ2 ),

where h11 , h12 , h22 are the coefficients of the second fundamental form, and O(λ2 ) means λ2 times some continuous function. So we have 2

λ λ λ g11 g22 − g12 = a0 + a1 λ + O(λ2 ), 2 where a0 = g11 g22 − g12 , a1 = −2h(g11 h22 + g22 h11 − 2g12 h12 ). This implies that for (u, v) ∈ △,

󵄨󵄨 󵄨󵄨 a1 󵄨󵄨 λ λ 󵄨 λ 2 1/2 λ)󵄨󵄨󵄨 < C1 λ2 , 󵄨󵄨(E G − F ) − (√a0 + 󵄨󵄨 2√a0 󵄨󵄨 where C1 > 0 is a constant. By the formulas 2 1/2

λ λ λ Aλ = ∬(g11 g22 − g12 ) dudv △

and 1/2

2 A0 = ∬(g11 g22 − g12 ) dudv = ∬ √a0 dudv △

we get

△

50 � 3 Minimal surfaces in ℝ3 󵄨󵄨 󵄨󵄨 a 󵄨 󵄨󵄨 2 󵄨󵄨Aλ − A0 − λ ∬ 1 dudv󵄨󵄨󵄨 < C1 λ . 󵄨󵄨 󵄨󵄨 2√a0 △

This implies that 󵄨󵄨 A − A 󵄨󵄨 a 󵄨󵄨 λ 󵄨 0 − ∬ 1 dudv󵄨󵄨󵄨 < C1 λ. 󵄨󵄨 󵄨󵄨 λ 󵄨󵄨 2√a0 △

Letting λ go to zero and using H =

1 g11 h22 −2g12 h12 +g22 h11 , 2 2 g11 g22 −g12

we get 1/2

2 A′ (0) = −2 ∬ hH(g11 g22 − g12 ) hdudv. △

This proves the theorem. Theorem 3.1.3. Let x : U → ℝ3 be a regular parameterized surface, and let D ⊂ U be a bounded domain in U. Then x is minimal on D if and only if A′ (0) = 0 for all such D and all normal variations of x(D). Proof. If x is minimal, then H is identically zero, and so A′ (0) = 0 for any h. Conversely, assume that A′ (0) = 0 for any h but that there exists q ∈ D for which its mean curvature H(q) ≠ 0. We choose h : D → ℝ such that h(q) = H(q) and h is identically zero outside a small neighborhood of q. Then A′ (0) < 0 for the variation determined by this h. This contradiction shows that H(q) = 0. Since q is arbitrary, x is minimal. Note that we have said nothing about the second derivative of A at 0, so that a minimal surface, although a critical point of A, may not actually be a minimum.

3.2 The Weierstrass–Enneper representation Let M be a surface in ℝ3 with parameterization x : U → ℝ3 . The parameterization x is said to be isothermal if ‖xu ‖ = ‖xv ‖

and xu ⋅ xv = 0.

Thus x is a conformal mapping (i. e., it preserves angles), and its induced metric from the standard Euclidean metric in ℝ3 is ds2 = λ2 (du2 + dv2 ), where λ = ‖xu ‖ = ‖xv ‖. Lemma 3.2.1 (Existence of isothermal parameters on a minimal surface). Let M be a minimal surface in ℝ3 . Then there is a (local) isothermal parameterization of M. Proof. Since it is a local problem, we can assume that M is the graph of some function, so we have a local parameterization of σ(x, y) = (x, y, f (x, y)) : U ⊂ ℝ2 → ℝ3 for M. The coefficients of the first fundamental form are

3.2 The Weierstrass–Enneper representation

g11 = 1 + fx2 ,

� 51

g22 = 1 + fy2 .

g12 = fx fy ,

The graph z = f (x, y) is a minimal surface if and only if H = 0, which means (1 + fy2 )fxx − 2fx fy fxy + (1 + fx2 )fyy = 0.

(3.1)

2 = √1 + f 2 + f 2 . It can easily be checked through a direct computaLet A = √g11 g22 − g12 x y tion by using (3.1) that

𝜕 g 𝜕 g12 ( ) = ( 11 ) 𝜕x A 𝜕y A

(3.2)

𝜕 g 𝜕 g22 ( ) = ( 12 ). 𝜕x A 𝜕y A

(3.3)

For example, to verify (3.2), we proceed as follows: fx fy 1 + fx2 𝜕 𝜕 ( )− ( ) 𝜕x √1 + f 2 + f 2 𝜕y √1 + f 2 + f 2 x y x y =

(1 + fx2 + fy2 )(fxx fy + fx fxy ) − fx fy (fx fxx + fy fyx ) (1 + fx2 + fy2 )3/2

− =

2(1 + fx2 + fy2 )fx fxy − (1 + fx2 )(fx fxy + fy fyy ) (1 + fx2 + fy2 )3/2

fy [(1 + fy2 )fxx − 2fx fy fxy + (1 + fx2 )fyy ] (1 + fx2 + fy2 )3/2

= 0.

Thus there are smooth functions ϕ and ψ on U ⊂ ℝ2 such that ϕx =

g11 , A

ϕy =

g12 , A

ψx =

g12 , A

ψy =

g22 . A

Set L(x, y) = (x + ϕ(x, y), y + ψ(x, y)) = (u, v) : U ⊂ ℝ2 → ℝ2 (here L is called the Levy map). Then the Jacobian matrix of L is J =(

1+

g11 A

g12 A

1

g12 A + gA22

),

(3.4)

and hence det(J) = 2 +

g11 + g22 > 0. A

̃ v) := Therefore by the inverse function theorem the inverse map L−1 exists. Let σ(u, (σ ∘ L−1 )(u, v). By the chain rule,

52 � 3 Minimal surfaces in ℝ3

(

x σ̃ u )=( u σ̃ v xv

yu σ )( x ). yv σy

Hence t

(

g σ̃ σ̃ u ) ⋅ ( u ) = J −1 ( 11 σ̃ v g12 σ̃ v

g12 1 A t ) (J −1 ) = ( g22 0 det J

0 ). 1

Hence we get σ̃ u ⋅ σ̃ u = σ̃ v ⋅ σ̃ v =

A , det J

σ̃ u ⋅ σ̃ v = 0.

(3.5)

This shows that the new parameterization σ̃ is isothermal. Let M be a regular minimal surface in ℝ3 of class C 2 . By the theorem above each point p ∈ M has a neighborhood in which an isothermal parameter x(u, v) is defined. Proposition 3.2.2. For an oriented surface M with metric ds2 , if we take two systems of positively oriented isothermal local parameters (or coordinates (u, v) and (x, y)), then w := u + √−1v is a holomorphic function in z := x + √−1y on the common domain of definition. Proof. By assumption there exists a positive differentiable function λ such that ds2 = λ2 (dx 2 + dy2 ). Therefore we have A := ux2 + v2x = uy2 + v2y ,

ux uy + vx vy = 0.

This means that the Jacobi matrix J := (

ux uy

vx ) vy

satisfies the identity J t J = A2 I2 , where I2 is the 2 × 2 identity matrix. We then have J −1 =

v 1 ( y −uy det J

−vx v 1 )= ( y ux A −uy

−vx ). ux

On the other hand, since (det J)2 = det(J t J) = det(A2 I2 ) = A2 and det J > 0, we have A = det J. This implies that ux = uy and uy = −vx . Therefore the function w = u + iv is holomorphic in z. From the above proposition assume that M is minimal and orientable. Then M is a Riemann surface with a family of local isothermal parameters that covers M, since such changes of coordinates are holomorphic with z = u + √−1v. Let x : M → ℝ3 be

3.2 The Weierstrass–Enneper representation

� 53

a minimal immersion, and let {U, ϕ = (u, v)} be a local isothermal coordinate. Denote x(u, v) = (x ∘ ϕ−1 )(u, v), which is a local isothermal parameterization. The evaluation of many quantities in the study of surfaces are simplified when considered in reference to the Riemann surfaces. For example, the induced metric is ds2 = λ2 (du2 + dv2 ) = λ2 dzd z,̄ where z = u + √−1v. We denote by △, or more precisely △z , the Laplacian operator 𝜕2 𝜕2 + 𝜕v 2 in terms of the holomorphic coordinate z = u + iv. If we take another local coor𝜕u2

dinate ζ , then we have △ζ = |dz/dζ |2 △z . Thus the operator △ds2 := (1/λ2 )△z is globally defined (independent of the choice of the holomorphic coordinate z) and is called the Laplacian operator with respect to the metric ds2 . Denote △ds2 :=

1 𝜕 𝜕 4 𝜕2 ( + ) = . λ2 𝜕u2 𝜕v2 λ2 𝜕z𝜕z̄

The Gauss curvature by (2.7) is K = − △ds2 log λ.

(3.6)

For an immersion x = (x1 , x2 , x3 ) : M → ℝ3 , we define △ds2 x = (△ds2 x1 , △ds2 x2 , △ds2 x3 ). We have △ds2 x = 2Hn.

(3.7)

To prove this identity, notice that x(u, v) is isothermal, so xu ⋅ xu = xv ⋅ xv and xu ⋅ xv = 0. By differentiating these equations with respect to u and v separately we get xuu ⋅ xu = xuv ⋅ xv

and

xvu ⋅ xv = −xvv ⋅ xu .

Hence (xuu + xvv ) ⋅ xu = xuv ⋅ xv − xvu ⋅ xv = 0. Similarly, (xuu + xvv ) ⋅ xv = 0. From this we conclude that △ds2 x is proportional to n. Now write nu = a11 xu + a21 xv and nv = a12 xu + a22 xv . Since H = 21 trace(SP ) = − 21 trace(dn|P ), we have 1 H = − (a11 + a22 ). 2 Hence λ2 (△ds2 x) ⋅ n = (xuu + xvv ) ⋅ n

= −xu ⋅ nu − xv ⋅ nv = −a11 ‖xu ‖ − a22 ‖xv ‖

54 � 3 Minimal surfaces in ℝ3 = −(a11 + a22 )λ2 = 2Hλ2 . This proves (3.7). Therefore we get the following proposition. Proposition 3.2.3. A surface M immersed in ℝ3 is minimal if only if its isothermal parameterization x is harmonic. Corollary 3.2.4. A minimal surface immersed in ℝ3 cannot be compact. Proof. If M were compact, then each xk , 1 ≤ k ≤ 3, would attain its maximum on M, where x = (x1 , x2 , x3 ) : M → ℝ3 is an immersion of M. Since by Proposition 3.2.3 each xk , 1 ≤ k ≤ 3, is harmonic, it would be constant, contradicting the assumption that M is a surface. Let M be a minimal surface immersed in ℝ3 with an immersion x : M → ℝ3 . We know that M is always noncompact. Take a local isothermal coordinate {U, ϕ = (u, v)} of M. Write x(u, v) = (x ∘ ϕ−1 )(u, v) = (x1 (u, v), x2 (u, v), x3 (u, v)). Let ϕ = (ϕ1 , ϕ2 , ϕ3 ), where, for 1 ≤ k ≤ 3, ϕk =

𝜕xk 𝜕x 1 𝜕x = ( k − √−1 k ). 𝜕z 2 𝜕u 𝜕v

The condition of M being minimal implies that x is harmonic, that is, 𝜕ϕk 𝜕z̄

= 0, so ϕk is holomorphic for k = 1, 2, 3. Also, 3

3

k=1

k=1

4 ∑ ϕ2k = ∑ (

2

𝜕2 x 𝜕z𝜕z̄

= 0. Thus

2

3 3 𝜕x 𝜕x 𝜕xk 𝜕x ) − ∑ ( k ) − 2√−1 ∑ k k 𝜕u 𝜕v 𝜕u 𝜕v k=1 k=1

= ‖xu ‖2 − ‖xv ‖2 − 2√−1xu ⋅ xv = 0,

and 3

3

k=1

k=1

4 ∑ |ϕk |2 = ∑ (

2

2

3 𝜕xk 𝜕x ) + ∑ ( k ) = ‖xu ‖2 + ‖xv ‖2 ≠ 0 𝜕u 𝜕v k=1

if M is regular. The differential αk := ϕk dz, 1 ≤ k ≤ 3, is independent of the local coordinate z and hence is a globally defined differential form on M. Thus we obtain the following: Proposition 3.2.5. Let M be a Riemann surface. Then α = ϕdz is a vector-valued holomorphic form on M if and only if x is a minimal surface. Moreover, x = 2 Re(∫ ϕ), where the integral is taken along any path from a fixed point to z on M.

3.2 The Weierstrass–Enneper representation

� 55

When the real part of the integral of α along any closed path is zero, we say that α has no real periods. The nonexistence of real periods for α is easily seen to be equivalent z to the independence of (Re ∫ α) on the path M. Theorem 3.2.6 (Weierstrass–Enneper representation). Let α1 , α2 , and α3 be holomorphic differentials on M such that αi = ϕi dz. Suppose that 1. ϕ21 + ϕ22 + ϕ23 = 0; 2.

3.

∑3k=1 |ϕk (z)|2 > 0;

Each αk has no real periods on M. z

Then the mapping x : M → ℝ3 defined by xk (z) = Re(∫ αk ) is a minimal immersion. z

Condition (3) of the theorem is necessary to guarantee that Re(∫z αk ) depends only 0 on the final point z. Therefore each xk is well defined independently of the path from z0 to z. It is obvious that ϕ = 𝜕x is holomorphic, and so x is harmonic. Hence x is a minimal 𝜕z surface. Condition (2) guarantees that x is regular, so it is an immersion. It is possible to give a simple description of all solutions of the equation ∑3k=1 ϕk = 0 on M. To do this, we assume that ϕ1 ≠ iϕ2 . (If ϕ1 = iϕ2 , then ϕ3 = 0, so that the minimal surface is a plane.) Let us define the holomorphic form ω and the meromorphic function g as follows: ω = α1 − iα2

and

g=

α3 . α1 − iα2 ϕ

3 Locally, write αk = ϕk dz. Then ω = fdz, where f = ϕ1 − iϕ2 and g := ϕ −iϕ . Then f is a 1 2 (locally defined) holomorphic function, and g is meromorphic on M, which is globally defined. We call the pair {ω, g} (or simply {f , g}) the Weierstrass–Enneper representation of the minimal surface M. The induced metric on the surface M is

2

ds2 = 2‖ϕ‖2 dzd z̄ = | f |2 (1 + |g|2 ) dzd z,̄

(3.8)

and its Gauss curvature by (3.6) is K =−

2|g ′ | . | f |2 (1 + |g|2 )4

(3.9)

Note that if z0 is a point where g has a pole of order m. Then it is clear that ω (or f ) must have a zero of order exactly 2m0 at z0 for each αk to be holomorphic. In terms of g and ω, we have 1 α1 = (1 − g 2 )ω, 2

i α2 = (1 + g 2 )ω, 2

α3 = gω.

Therefore the minimal immersion x = (x1 , x2 , x3 ) is given by

56 � 3 Minimal surfaces in ℝ3 z

x1 (z) = Re ∫ f (1 − g 2 )dζ , z0

z

x2 (z) = Re ∫ if (1 + g 2 )dζ , z0

z

x3 (z) = 2 Re ∫ fgdζ , z0

where the integral is taken along an arbitrary path from the fixed point z0 ∈ D to the point z. Conversely, let D ⊂ ℂ be a simply connected domain, let f be a holomorphic function in D, and let g be a meromorphic function in D such that fg 2 is holomorphic in D. Let 1 ϕ1 = f (1 − g 2 ), 2

i ϕ2 = f (1 + g 2 ), 2

ϕ3 = fg.

Then it is clear that ϕ1 , ϕ2 , ϕ3 are holomorphic and 1 1 2 2 ϕ21 + ϕ22 + ϕ23 = f 2 (1 − g 2 ) − f 2 (1 + g 2 ) + f 2 g 2 4 4 1 1 = f 2 (1 − 2g 2 + g 4 ) − f 2 (1 + 2g 2 + g 4 ) + f 2 + g 2 4 4 = −f 2 g 2 + f 2 g 2 = 0. This gives an alternative version of the Weierstrass–Enneper representation. Theorem 3.2.7 (Weierstrass–Enneper representation). If D is a simply connected domain in the complex plane and if f is a holomorphic function in D and g is a meromorphic function in D such that fg 2 is holomorphic in D, then the minimal surface is defined by the parameterization x(z) = (x1 (z), x2 (z), x3 (z)) where z

x1 (z) = Re ∫ f (1 − g 2 )dζ , z0

z

x2 (z) = Re ∫ if (1 + g 2 )dζ , z0

z

x3 (z) = 2 Re ∫ fgdζ , z0

where the integral is taken along an arbitrary path from the fixed point z0 ∈ D to the point z.

3.2 The Weierstrass–Enneper representation

� 57

It can be shown that the catenoid and helicoid have representations of the forms (f (z), g(z)) = (− 21 e−z , −ez ) and (f (z), g(z)) = (− 2i e−z , −ez ), respectively. It can also be shown that the minimal surface called the Enneper surface has a representation of the form (f (z), g(z)) = (1, z). The function g has a special geometric meaning: g is the stereographic projection of n (the unit normal to the minimal surface) from the unit sphere to ℂ ∪ {∞} = ℙ1 (ℂ). Here the stereographic projection from the north pole is the mapping St : S 2 \{(0, 0, 1)} → ℝ2 defined by St(x1 , x2 , x3 ) = (

x1 x2 , , 0). (1 − x3 ) (1 − x3 )

Note that the stereographic projection from the north pole is a bijective, smooth, and conformal mapping where it is defined. We now identify ℝ2 with ℂ and extend the map St to a bijective mapping St : S 2 → ℂ ∪ {∞} defined by (x1 , x2 , x3 ) 󳨃→

x1 + √−1x2 1 − x3

and (0, 0, 0) 󳨃→ ∞.

(3.10)

Theorem 3.2.8. Let M be a minimal surface with isothermal parameterization x(u, v) and Weierstrass–Enneper representation (f , g). Then g = St(n), that is, g : M → ℙ1 (ℂ) can be regarded as the Gauss map of M given by n : M → S 2 . Proof. Let x = (x1 , x2 , x3 ) be a parameterization of M. To compute n, we first compute xu × xv . By a direct computation, xu × xv = ((x2 )u (x3 )v − (x3 )u (x2 )v , (x3 )u (x1 )v − (x1 )u (x3 )v , (x1 )u (x2 )v − (x2 )u (x1 )v ). Notice that (x2 )u (x3 )v − (x3 )u (x2 )v = Im[((x2 )u − i(x2 )v )((x3 )u + i(x3 )v )] ̄ = Im[2(𝜕x2 /𝜕z) ⋅ 2(𝜕x3 /𝜕z)] = 4 Im(ϕ ϕ̄ ). 2 3

Similarly, (x3 )u (x1 )v − (x1 )u (x3 )v = 4 Im(ϕ3 ϕ̄ 1 ) and (x1 )u (x2 )v − (x2 )u (x1 )v = 4 Im(ϕ1 ϕ̄ 2 ). Hence xu × xv = 4 Im(ϕ2 ϕ̄ 3 , ϕ3 ϕ̄ 1 , ϕ1 ϕ̄ 2 ) = | f |2 (1 + |g|2 )(2 Re(g), 2 Im(g), (|g|2 − 1)),

58 � 3 Minimal surfaces in ℝ3 where we used z − z̄ = 2 Im z. Since x(u, v) is isothermal, ‖xu × xv ‖ = ‖xu ‖‖xv ‖ = ‖xu ‖2 = 2‖ϕ‖2 = |f |2 (1 + |g|2 )2 , it follows that n=

xu × xv 2 Re(g) 2 Im(g) |g|2 − 1 =( , , ). ‖xu × xv ‖ 1 + |g|2 1 + |g|2 1 + |g|2

Hence from (3.10) we have St(n) =

2 Re(g) 1+|g|2

Im(g) + √−1 21+|g| 2

1−

|g|2 −1 1+|g|2

= g.

3.3 Gauss maps of complete minimal surfaces in ℝ3 In this section, we prove the Bernstein’s theorem and its extensions. For the references, see [4, 5, 6, 14, 18, 27, 30, 31, 32, 33, 42, 49, 55, 56, 57, 58, 59, 60, 61, 62, 87, 88, 89]. To do so, we need the following result regarding the flat metric on a Riemann surface. Lemma 3.3.1. Let dσ 2 be a conformal flat metric on an open surface M (i. e., its curvature is identically zero). Then for each point p ∈ M, there exists a local diffeomorphism Φ of a disk D(R) = {w ∈ ℂ | |w| < R}(0 < R ≤ ∞) onto an open neighborhood of p in M with Φ(0) = p such that Φ is a local isometry (i. e., the pullback Φ∗ (dσ 2 ) is equal to the standard Euclidean metric dsE2 on D(R) ⊂ ℂ), and there exists a point a0 with |a0 | = 1 such that the Φ-image Γa0 of the line La0 = {w = a0 t : 0 < t < R} is divergent in M. In particular, if the metric dσ 2 is flat and complete, then M is biholomorphically isometric to the complex plane ℂ with its standard flat metric |dw|2 . Proof. Write dσ 2 = |f (z)|2 |dz|2 . Then f is holomorphic and without zeros. Define z

w = F(z) = ∫ f (ζ )dζ , p

z

where the integral ∫p f (ζ )dζ is over a simple path on M with initial point p and endpoint z. Since f is holomorphic, the integral is independent of the choices of the paths, | ≠ 0, F maps an open neighborhood U of p biand hence F is well defined. Since dw dz p holomorphically onto an open disc D(R) := {w ∈ ℂ | |w| < R}(0 < R ≤ ∞). For the case R = ∞, we have M = U. In this case, every curve Γa0 is divergent. So we assume that R < ∞. Let R0 be the least upper bound of R > 0 such that F biholomorphically maps some open neighborhood of p onto D(R). By definition there is a sequence {Rn } that converges to R0 such that F|Un : Un → D(Rn ) is biholomorphic from some neighborhood Un of p. Then F maps U0 = ⋃∞ n=1 Un onto D(R0 ). Consider the inverse map Φ : D(R0 ) → U0 . Assume that Γa0 is not divergent in M for some a0 with |a0 | = 1. Then there is a sequence {tn } with 0 ≤ tn < R0 such that limn→∞ tn = R0 and ϕ(tn a0 ) converges to a point q ∈ M.

3.3 Gauss maps of complete minimal surfaces in ℝ3

� 59

Since F(q) ≠ 0, F biholomorphically maps some open neighborhood of q onto an open neighborhood of w0 := R0 a0 . Therefore Φ is biholomorphically extended to a neighborhood of w0 . If there exists no curve Γa0 that diverges in M for all a0 with |a0 | = 1, then Φ is biholomorphically extended to D(R) for some R > R0 . This contradicts with the choice of R0 . Therefore some Γa0 diverges in M. Finally, we show that Φ is an isometry. Indeed, 󵄨 󵄨2 󵄨2 󵄨󵄨 dz 󵄨󵄨󵄨 󵄨 2 2 Φ∗ (dσ 2 ) = 󵄨󵄨󵄨(f ∘ Φ)󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 |dw| = |dw| . 󵄨󵄨 dw 󵄨󵄨󵄨 Since 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨−1 󵄨󵄨 󵄨=󵄨 󵄨 = 󵄨󵄨󵄨(f ∘ Φ)󵄨󵄨󵄨 , 󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 Φ is an isometry. Theorem 3.3.2 (Bernstein). Let M ⊂ ℝ3 be the graph z = f (x, y). If M is minimal and f is defined on the whole plane ℝ2 , then f (x, y) = ax + by + c (i. e., M is a plane). Proof. Let x(x, y) = (x, y, f (x, y)) : ℝ2 → ℝ3 be the standard parameterization of M. Take ̃ v) = (x∘L−1 )(u, v) as described in Lemma 3.2.1, the new isothermal parameterization x(u, where L is the Levy map. The induced metric on M can be written as ds2 = λ2 (du2 + dv2 ), where, according to (3.5), λ2 =

1 + fx2 + fy2 2√1 + fx2 + fy2 + 2 + fx2 + fy2

=

A2 A2 = 2A + A2 + 1 (A + 1)2

(3.11)

with A = √1 + fx2 + fy2 . Its Gauss curvature is, by formula (3.6), K=

−1 𝜕2 𝜕2 1 ( 2 + 2 ) log λ = − △ds2 log λ = △ds2 log(1 + ). 2 A λ 𝜕u 𝜕v

(3.12)

We define the new metric dσ 2 = (1 +

2

1 ) ds2 = (du2 + dv2 ). A

Then dσ 2 is a flat metric. Also, since ds ≤ dσ ≤ 2ds, we know that the metric dσ 2 on M is complete since ds2 is complete. By Lemma 3.3.1, M with the metric dσ 2 is isometric to the complex plane ℂ with its standard flat metric d|w|. On the other hand, since K ≤ 0, by (3.12) the function ψ := −log(1 + A1 ), considered as a function on the ℂ-plane, is

60 � 3 Minimal surfaces in ℝ3 subharmonic. Clearly, ψ ≤ 0. By the well-known “Hadamard three-circle theorem” for subharmonic functions, for any r1 < r < r2 , noticing that max|x|=r2 ψ(x) ≤ 0, max ψ(x) ≤ |x|=r

r 1 r (max ψ(x) log 2 + max ψ(x) log ) r |x|=r2 r1 log rr2 |x|=r1 1

r2 1 ≤ . r2 max ψ(x) log r log r |x|=r1 1

By letting r2 → ∞, we get, for any z ∈ ℂ with 0 < r1 < |z|, ψ(z) ≤ max|x|=r1 ψ(x). Because ψ is upper semicontinuous, this implies that ψ(z) ≤ lim sup max ψ(x) = ψ(0), r1 →0

|x|=r1

that is, ψ has a maximum at z = 0 and therefore is constant by the maximum principle for subharmonic functions. Thus K ≡ 0 by (3.12), so the surface must be a plane. The Bernstein theorem was extended by Heinz [41] in 1952 to get the following Gauss curvature estimate. Its proof is omitted here. Theorem 3.3.3 ([41]). Let M be a minimal graph over a disk of radius R. Then |K(0)| ≤ c/R2 , where c is a universal constant. The graph of a function z = f (x, y) is a very particular surface. Such a surface passes a “vertical line test”, that is, no two points on the surface will lie on a given vertical line. In other words, the image of its Gauss map is contained in either the upper hemisphere or the lower hemisphere. Therefore the image of its Gauss map always omits half of the unit sphere. For a general (not necessarily the graph of some function) complete nonplanaer minimal surface, Nirenberg took the point of view about the image of the Gauss map and made the following conjecture. Nirenberg’s conjecture. A complete simply connected minimal surface must be a plane if the normals 1. omit a neighborhood of some direction or 2. omit more than two directions (compare with the little Picard theorem). R. Osserman in 1959–1965 showed that the Nirenberg conjecture 1 is true and 2 is false: The Gauss map to the classical Scherk surface omits four points. Xavier [88] in 1981 showed that the unit normals of a nonflat complete minimal surface in ℝ3 cannot omit seven directions. Fujimoto [27] in 1988 solved the long-standing conjecture: The normals to a complete nonplanar minimal surface cannot omit more than four directions. The finite version (bound of the Gauss curvature) is also obtained. Below we present their proofs. We first define the meaning of completeness. Let M be a surface. A continuous curve γ(t), 0 ≤ t ≤ 1, in M is said to be divergent in M if for each compact set G, there is t0 such

3.3 Gauss maps of complete minimal surfaces in ℝ3

�

61

that γ(t) ∈ ̸ G for any t ≥ t0 . We define the distance d(p)(≤ +∞) from a point p ∈ M to the boundary as the greatest lower bound of the lengths of all continuous curves that are divergent in M. A surface M is said to be complete if d(p) = ∞ for all p ∈ M. Theorem 3.3.4 (Osserman [56, 59]). Let M be a complete minimal surface immersed in ℝ3 . If its Gauss map is not dense, then M is a plane. Proof. Let x = (x1 , x2 , x3 ) : M → ℝ3 be an immersion. Take a local isothermal coordinate 𝜕x (U, ϕ) for M, and let x(u, v) = (x ∘ ϕ−1 )(u, v). Let ϕk = 𝜕zk for z = u + iv. Then ϕk is

holomorphic for k = 1, 2, 3. Let f = ϕ1 − iϕ2 and g = 2

2

2 2

2

ϕ3 ; ϕ1 −iϕ2

then the induced metric is

ds = |f | (1+|g| ) |dz| . As we know, g is the Gauss map. So, assuming that the Gauss map of M is not dense, we can assume that g omits ∞ and there exists C > 0 such that |g| < C. Thus ds2 = |f |2 (1 + |g|2 )2 |dz|2 ≤ (1 + C)|f |2 |dz|2 , and hence the flat metric d s2̄ := |f |2 |dz|2 is also complete. By Lemma 3.3.1, M with the flat metric d s2̄ := |f |2 |dz|2 is isometric to the complex plane ℂ. Therefore g, which is bounded, is a holomorphic function on the whole complex plane. By Liouville’s theorem, g is constant, and thus Kds2 ≡ 0. Hence the surface must be a plane by Theorem 3.1.1. Theorem 3.3.5 (Fujimoto, [27]). Let M be a complete minimal surface immersed in ℝ3 . If its Gauss map omits more than four values in S 2 , then M lies in a plane. R

Proof. The proof is similar to the above, except that instead of using the fact ∫0 0 |dw| = 1

a0 R0 < ∞, we use the fact that ∫0 dt < ∞ for any 0 < p < 1. We also need to use tp 2 Corollary 2.3.9 to estimate (1 + |g| ) since in our case, |g| < C is no longer true. By lifting g to the universal cover of M, we can assume that M is simply connected. So M is biholomorphic to either ℂ or the unit-disc. If M = ℂ, then by the little Picard theorem, g is constant. So the surface is flat. Hence we can assume that M is bioholomorphic to the unit disc. The induced metric, according to (3.8), is 2

ds2 = | f |2 (1 + |g|2 ) |dz|2 . We assume that g omits a1 , . . . , aq ∈ ℂ ∪ {∞} with q ≥ 5 and aq = ∞. We also assume that g is not constant. Take a positive η such that q − 6 < qη < q − 4. Similarly to the proof of Theorem 3.3.4, we seek a flat metric d s2̄ on M, which allows us to get a local isometry w = Φ (or change of coordinate by letting w = Φ(z)) from the disk D(R) to M and a divergent path in M, as stated in Lemma 3.3.1. As in the proof of Lemma 3.3.1, the flat metric (or Φ) can be decided through computing dw/dz (see (3.16)). Note that 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 󵄨󵄨 dz 󵄨󵄨󵄨 Φ∗ ds = 󵄨󵄨󵄨(f ∘ Φ)(w)󵄨󵄨󵄨(1 + 󵄨󵄨󵄨(g ∘ Φ)(w)󵄨󵄨󵄨 )󵄨󵄨󵄨 󵄨|dw|. 󵄨󵄨 dw 󵄨󵄨󵄨 To estimate Φ∗ ds, we want

62 � 3 Minimal surfaces in ℝ3

Φ∗ ds = (

|h′ (w)|(1 + |h(w)|2 )1/p q−1

∏j=1 |h(w) − aj |1−η

p

) |dw|,

(3.13)

where p = 2/(q − 2 − qη) and h(w) = (g ∘ Φ)(w). Note that the purpose of doing so is to estimate Φ∗ ds by using Corollary 2.3.9 in Chapter 2 for h(w) = (g ∘ Φ)(w) on D(R), that is, |h′ |(1 + |h|2 )1/p

q−1 ∏j=1 |h

− aj

|1−η

≤C

R . R2 − |z|2

(3.14)

Notice that (3.13) is the same as p 󵄨 󵄨 |h′ (w)| 󵄨󵄨 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 ) . 󵄨󵄨(f ∘ Φ)(w)󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 = ( q−1 󵄨󵄨 dw 󵄨󵄨 ∏j=1 |h(w) − aj |1−η

Note that, by the chain rule, h′ =

dg dz , dz dw

(3.15)

and thus (3.15) becomes

p 󵄨 󵄨 || dz | | dg 󵄨󵄨 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 dz dw ) , 󵄨󵄨(f ∘ Φ)(w)󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 = ( q−1 󵄨󵄨 dw 󵄨󵄨 ∏j=1 |(g ∘ Φ)(w) − aj |1−η

that is, 󵄨󵄨 dw 󵄨󵄨1−p | f (z)| ∏q−1 |g(z) − aj |p(1−η) j=1 󵄨󵄨 󵄨󵄨 , 󵄨 = 󵄨󵄨 󵄨󵄨 dz 󵄨󵄨󵄨 |g ′ (z)|p or 1

󵄨󵄨 dw 󵄨󵄨 | f (z)| 1−p ∏q−1 |g(z) − aj | j=1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = p 󵄨󵄨 dz 󵄨󵄨 |g ′ (z)| 1−p

p(1−η) 1−p

(3.16)

.

This suggests the flat metric we desire. To carry out the proof, we thus define the following new flat metric (with zero Gauss curvature): 2

dσ 2 := C

p − 1−p

q−1

| f (z)| 1−p ∏j=1 |g(z) − aj | 2p

|g ′ (z)| 1−p

2p(1−η) 1−p

|dz|2

on M ′ = M − {g ′ = 0}. With the flat metric dσ 2 on M ′ , fix a point p0 ∈ M ′ . By Lemma 3.3.1 there exists a local diffeomorphism Φ of a disk D(R) = {w ∈ ℂ : |w| < R} (0 < R ≤ ∞) onto an open neighborhood of p0 with Φ(0) = p0 such that Φ is a local isometry. Since g is not constant, R < ∞ by the little Picard theorem. In addition, by Lemma 3.3.1 there exists a point a0 with |a0 | = 1 such that the Φ-image Γa0 of the curve La0 = {w = a0 t : 0 < t < R} is divergent in M ′ = M − {g ′ = 0}. From the above discussion, with the choice of our flat metric, (3.13) holds, that is,

3.3 Gauss maps of complete minimal surfaces in ℝ3

Φ∗ ds = (

|h′ (w)|(1 + |h(w)|2 )1/p q−1

∏j=1 |h(w) − aj |1−η

� 63

p

) |dw|.

We claim that the path Γa0 does not diverge to zeros of g ′ . Indeed, if Γa0 diverges to a point z0 such that g ′ (z0 ) = 0, then by taking a local coordinate on V , a neighborhood

of z0 , with ζ (z0 ) = 0, we can write dσ 2 = |ζ | 1−p β|dζ |2 with β > 0 on V . Thus, because p/(1 − p) > 1, −2p

R = ∫ dσ ≥ C ∫ Γa0

Γa0 ∩V

1

p

|ζ | 1−p

|dζ | = ∞,

which gives a contradiction. Hence the path diverges to the boundary M. Denote by length(Γa0 ) the length of the part Γa0 on M with respect to the induced metric ds2 . Then length(Γa0 ) = ∫ ds = ∫ Φ∗ ds Γa0

La0

R

= ∫( 0

|h′ (w)|(1 + |h(w)|2 )1/p q−1

∏j=1 |h(w) − aj |1−η

R

≤C ∫ ′

0

p

) |dw|

Rp dt < ∞, (R2 − t 2 )p

where p = 2/(q − 2 − qη) < 1 when q ≥ 5. This contradicts with the fact that the induced metric ds2 := |f |2 (1 + |g|2 )2 |dz|2 is complete. Therefore g must be constant, and thus the surface lies in a plane by Theorem 3.1.1. Actually, Fujimoto’s five-point theorem can be reformulated to the following “complex analysis” version (see Theorem 5.2.1 for the related results): Theorem 3.3.6 ([46]). Let M be an open Riemann surface, and let g be a meromorphic function on M. Consider the conformal metric on M given by m

ds2 = (1 + |g|2 ) |ω|2 , where ω is a holomorphic 1-form, and m ∈ ℤ≥0 . Assume that M is complete with respect to this metric. If g is nonconstant, then g can omit at most m + 2 distinct values in ℂ ∪ {∞}. The following is Xavier’s result [88]. Although it is weaker than that of Fujimoto, his proof uses the integral results, especially the logarithmic derivative lemma in Nevanlinna theory, which we will prove in the next chapter (Proposition 4.1.6). His method allows us to study maps from balls in ℂn into the complex projective spaces, which is an independent theory described in Chapter 6.

64 � 3 Minimal surfaces in ℝ3 Theorem 3.3.7 ([88]). Let M be a complete minimal surface immersed in ℝ3 . If its Gauss map omits more than seven values in S 2 , then M must be a plane. The proof of Theorem 3.3.7 is based on the following result of Yau [90]. We first prove a lemma. Lemma 3.3.8 ([90]). Let M n be a complete Riemannian manifold of dimension n, and let ω be a smooth integrable (n−1)-form defined on M n . Then there exists a sequence of domains {Bi } in M n such that Bi ⊂ Bi+1 , M n = ⋃i Bi , and limi→∞ ∫B dω = 0. i

n

Proof. Let r be the distance function on M to a (fixed) point p ∈ M n . Then r is a Lipschitz function on M n . By approximating the function r (cf. [35]) we can find a nonnegative smooth function gR such that (1) For all but a finite number of t < R, gR−1 (t) is a compact regular hypersurface; (2) |dgR | ≤ 3/2 on gG−1 ([0, R]); (3) gR−1 (t) ⊂ B(t + 1)\B(t − 1) for t ≤ R, where B(R) is a ball centered at p with radius R. On the other hand, Theorem 3.2.22 of [22] shows that R

|dgR ||ω| = ∫( ∫ |ω|)dt.

∫

0

gR−1 ([0,R])

gR−1 (t)

It follows from (2) that R

∫( ∫ |ω|)dt ≤ 0

3 ∫ |ω|. 2 M

gR−1 (t)

Therefore, for some R/2 ≤ tR ≤ R, where gR−1 (tR ) is a compact regular hypersurface, ∫ |ω| ≤

3 ∫ |ω|. R M

gR−1 (tR )

By Stokes’ theorem, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

∫

gR−1 ([0,tR ])

󵄨󵄨 󵄨 dω󵄨󵄨󵄨 ≤ 󵄨󵄨

∫ |ω| ≤ gR−1 (tR )

3 ∫ |ω|. R

This, together with property (3) of gR , we get M n = ⋃ gR−1 ([0, ti ]) i

M

3.3 Gauss maps of complete minimal surfaces in ℝ3

� 65

and lim

i→∞

∫

dω = 0.

gR−1 ([0,ti ])

Theorem 3.3.9 (Yau [90]). Let M be a complete Riemannian manifold of infinite volume. Let u be a nonnegative smooth function satisfying △ log u ≥ 0 almost everywhere. Then ∫M up dA = ∞ for any p > 0 unless u is a constant function. Proof. We only prove the case where p = 1 (the general case is essentially the same since △ log up = p △ log u ≥ 0). For every ϵ > 0, let vϵ = (u + ϵ)1/2 . Then △ log vϵ ≥

ϵ|du|2 ≥ 0, 2u(u + ϵ)2

that is, vϵ △ vϵ ≥ |dvϵ |2 .

(3.17)

Let r be the distance function on M to a (fixed) point p ∈ M. For arbitrary R1 , R2 such that 0 < R1 < R2 , let ω = ϕ(

r + R2 − 2R1 ), R2 − R1

where ϕ is a smooth function such that 0 ≤ ϕ ≤ 1, ϕ(t) = 1 for t ≤ 1, and ϕ(t) = 0 for t ≥ 2. Then ω is a continuous Lipschitz function, 0 ≤ ω ≤ 1, and ω(x) = 1 for x ∈ B(R1 ) and ω(x) = 0 for x ∈ M\B(R2 ). Also, |dω| ≤ C/(R2 − R1 ), where C > 0 is a constant. By the Stokes theorem, ∫ dvϵ ∧ ⋆d(ω2 vϵ ) = − ∫ ω2 vϵ △ vϵ . B(R2 )

B(R2 )

By (3.17) and (3.18), we have ∫ ω2 |dvϵ |2 ≤ − ∫ 2vϵ dvϵ ∧ ⋆ωdω − ∫ ω2 |dvϵ |2 . B(R2 )

B(R2 )

B(R2 )

Hence 2 ∫ ω2 |dvϵ |2 ≤ ∫ ω2 |dvϵ |2 + ∫ v2ϵ |dω|2 . B(R2 )

Since |dω| ≤ C/(R2 − R1 ), we see that

B(R2 )

B(R2 )

(3.18)

66 � 3 Minimal surfaces in ℝ3 1 ω2 |du|2 C2 = ∫ ω2 |dvϵ |2 ≤ ∫ ∫ v2ϵ . 4 u+ϵ (R2 − R1 )2 B(R2 )

B(R2 )

B(R2 )

By taking ϵ → 0 we get ω2 |du|2 C2 1 ≤ ∫ ∫ u. 4 u (R2 − R1 )2 B(R2 )

B(R2 )

We now prove the theorem by contradiction. Assume that ∫M udA < ∞ and u is not a constant function. By taking R2 = 2R1 and letting R1 → ∞ the above inequality becomes ∫ M

|du|2 < ∞. u

By the Schwarz inequality we get 2

(∫ |du|) ≤ ∫ M

M

|du|2 ∫ u < ∞. u M

Let uϵ = u + ϵ. Then △ log uϵ =

ϵ |du|2 . u(u + ϵ)2

Since ∫ |d log uϵ | = ∫ M

M

|du| |du| ≤∫ < ∞, u+ϵ ϵ M

if follows from Lemma 3.3.8 that there exists a sequence of domains {Bi } such that 0 = lim ∫ △ log uϵ = ∫ i→∞

Bi

M

ϵ |du|2 ≥ 0 u(u + ϵ)2

for all ϵ > 0. Hence ∫ M

|du|2 = 0. u(u + ϵ)2

This implies that u is constant, a contradiction. Hence ∫M udA = ∞. Lemma 3.3.10. Let g be a nowhere zero holomorphic function on the unit disk △(1). Assume that for 0 < r < 1, Tg (r) ≤ C log

1 , 1−r

3.3 Gauss maps of complete minimal surfaces in ℝ3

�

67

where Tg (r) is Nevanlinna’s characteristic function defined in Chapter 4 (see (4.8)). Then there exists a positive constant K such that for 0 < r < 1, 2π

K 1 󵄨󵄨󵄨󵄨 g ′ iθ 󵄨󵄨󵄨󵄨 1 log . ∫ 󵄨󵄨 (re )󵄨󵄨dθ ≤ 󵄨 󵄨 2π 󵄨 g 1−r 1−r 󵄨 0

Proof. By the assumption, log |g| is harmonic on △(1). For a given z in D(R) with R < 1, 2 we consider the linear transformation L(w) = RR2(z−w) , which sends z to zero and satisfies ̄ −zw |L(w)| = R if |w| = R. Let G(w) = log |g(L(w))|. Applying Theorem 2.2.1 (mean value theorem) for harmonic functions to G(w), we have 2π

dw dθ 󵄨 󵄨 log󵄨󵄨󵄨g(z)󵄨󵄨󵄨 = G(0) = ∫ log G(Reiθ ) = ∫ G(w) . 2π 2πiw 0

|w|=R

We let ζ = L(w). Then w = L−1 (ζ ) =

R2 (z − ζ ) . R2 − zζ̄

So, for |ζ | = R, dw −1 1 −1 dζ z̄ z̄ )dζ = ( = ( + 2 + ) ̄ ̄ 2πiw 2πi z − ζ R − zζ z − ζ ζ ζ − zζ̄ 2πi 2 ̄ R − |z|2 dζ −ζ z dζ + = . =( ) z − ζ ζ ̄ − z̄ 2πiζ |ζ − z|2 2πiζ Note that when |w| = R, |ζ | = R, and

dζ iζ

= dθ, we get the following Poisson formula:

2π

2 2 󵄨 󵄨 󵄨 󵄨 R − |z| dθ log󵄨󵄨󵄨g(z)󵄨󵄨󵄨 = ∫ log󵄨󵄨󵄨g(Reiθ )󵄨󵄨󵄨 . | Reiθ −z|2 2π 0

As Re(

ζ +z |ζ |2 − |z|2 )= , ζ −z |ζ − z|2

using the fact that 󵄨 󵄨 1 log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = [log f (z) + log f (z)] 2 and (log f (z))′ = 0, and thus

(3.19)

68 � 3 Minimal surfaces in ℝ3 f ′ (z) ′ 󵄨′ 󵄨 = (log f (z)) = 2(log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨) f (z) for any nowhere zero holomorphic function f , by differentiating both sides of (3.19), we get 2π

(

iϕ g′ dϕ 󵄨 Re 󵄨 )(z) = 2 ∫ log󵄨󵄨󵄨g(Reiϕ )󵄨󵄨󵄨 . iϕ g (Re −z)2 2π

(3.20)

0

On the other hand, by Corollary 4.1.2, 2π

∫ 0

dθ

2π

| Reiϕ −reiθ |2

=∫ 0

dθ 2π = . |R − reiθ |2 R2 − r 2

Since | log |x|| = log+ x + log+ (1/x) for x > 0, by the first main theorem (see Chapter 4) we have 2π

󵄨 󵄨 󵄨󵄨 dϕ = mg (R, ∞) + mg (R, 0) ≤ 2Tg (R) + O(1). ∫ 󵄨󵄨󵄨log󵄨󵄨󵄨g(Reiϕ )󵄨󵄨󵄨󵄨󵄨󵄨 2π 0

1 By the assumption Tg (R) ≤ C log 1−R we conclude that 2π 1 󵄨󵄨󵄨󵄨 g ′ iθ 󵄨󵄨󵄨󵄨 1 C′ log ∫ 󵄨󵄨 (re )󵄨󵄨dθ ≤ 󵄨󵄨 2π 󵄨󵄨 g (R − r) 1−R 0

for some constant C ′ > 0. By taking R = (1 + r)/2 we obtain the desired inequality. Proof of Theorem 3.3.7. By lifting g to the universal cover of M we can assume that M is simply connected. So M is biholomorphic to either ℂ or the unit disc. If M = ℂ, then by Picard’s theorem, g is constant. So the surface is flat. Without loss of generality, we assume that M = △(1), the unit disc. By applying Theorem 4.1.10 (see Chapter 4) the Tg (r) = ∞. little Picard theorem still holds for g : △(1) → ℙ1 = ℂ ∪ {∞} if lim supr→1 log(1−r) Hence we still can get that g is constant (so the surface is flat) under the assumption that Tg (r) lim supr→1 log(1−r) = ∞. It remains to prove the case that Tg (r) ≤ C log

1 . 1−r

Assume that g omits ∞ and 0. Then, by Lemma 3.3.10, 2π 1 󵄨󵄨󵄨󵄨 g ′ iθ 󵄨󵄨󵄨󵄨 K 1 log ∫ 󵄨 (re )󵄨󵄨dθ ≤ 󵄨󵄨 2π 󵄨󵄨󵄨 g 1−r 1−r 0

(3.21)

3.3 Gauss maps of complete minimal surfaces in ℝ3

� 69

for some constant K > 0. We claim that on △(1) we have the following inequality: |g ′ |5/6 (1 + |g|2 )2 ∏6j=1 |g − aj |5/6

6

≤C∑ i=1

6 󵄨󵄨 (g − a )′ 󵄨󵄨5/6 |g ′ |5/6 j 󵄨󵄨 󵄨󵄨 = C ∑ 󵄨󵄨 . 󵄨󵄨 5/6 󵄨 (g − a |g − aj | j ) 󵄨󵄨 i=1 󵄨

Indeed, let δ = mini=j̸ {|ai − aj |, 1}. For z, let i0 be the index among {1, . . . , 6} such that |g(z) − ai0 | ≤ |g(z) − aj | for 1 ≤ j ≤ 6. Then, for j ≠ i0 , by the triangle inequality 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 δ ≤ |ai0 − aj | ≤ 󵄨󵄨󵄨g(z) − ai0 󵄨󵄨󵄨 + 󵄨󵄨󵄨g(z) − aj 󵄨󵄨󵄨 ≤ 2󵄨󵄨󵄨g(z) − aj 󵄨󵄨󵄨, and hence |g(z) − aj | > δ/2. Thus |g ′ (z)|5/6 (1 + |g(z)|2 )2 ∏6j=1 |g(z) − aj |5/6

≤ K1

|g ′ (z)|5/6 (1 + |g(z)|2 )2 |g(z) − ai0 |5/6

for some constant K1 > 0. If |g(z) − ai0 | ≤ 1/3, then g is bounded, so the claim holds. If

|g(z) − ai0 | > 1/3 (so |g(z) − aj | > 1/3 for all 1 ≤ j ≤ 6), then and hence the claim still holds. This proves the claim. Thus by (3.21) we get 2π

∫( 0

|g ′ |5/6 (1 + |g|2 )2 ∏6j=1 |g − aj |5/6

(1+|g(z)|2 )2 ∏6j=1,j=i̸ |g(z)−aj |5/6 0

is bounded,

6 2π󵄨󵄨 (g − a )′ 󵄨󵄨󵄨5/6 j 󵄨 )(reiθ )dθ = C ∑ ∫ 󵄨󵄨󵄨 (reiθ )󵄨󵄨󵄨 dθ 󵄨 (g − aj ) 󵄨󵄨 i=1 󵄨 0

2π

5/6 󵄨󵄨 󵄨󵄨 (g − aj )′ 1 5/6 C ′′ 󵄨 󵄨 (log (reiθ )󵄨󵄨󵄨dθ) ≤ ) ≤ C ∑( ∫ 󵄨󵄨󵄨 5/6 󵄨󵄨 󵄨󵄨 (g − aj ) 1 − r (1 − r) i=1 6

′

0

for some positive constants C, C ′ , C ′′ . Note that the induced metric on △(1) is ds2 = |f |2 (1 + |g|2 )2 |dz|2 . Note that since g omits ∞, f is holomorphic without zeros. Take u := |g ′ | . Then 6 12/5 |f |

∏j=1 |g−aj |

|g ′ |5/6 (1 + |g|2 )2

∫ u5/6 dA = ∫ △(1)

△(1) 1

∏6j=1 |g − aj |5/6 2π

= ∫ rdr( ∫ ( 0

r

≤K∫ 0

0

dudv

|g ′ |5/6 (1 + |g|2 )2 ∏6j=1 |g

− aj

|5/6

)(reiθ )dθ) 5/6

r 1 (log ) 5/6 (1 − r) (1 − r)

dr < ∞.

So ∫△(1) up dA < ∞ for p = 5/6. Note that u satisfies △ log u = 0 almost everywhere, and also the minimal surface M has infinite volume (every complete simply connected

70 � 3 Minimal surfaces in ℝ3 surface with nonpositive curvature has infinite volume). Hence, by Theorem 3.3.9, ∫△(1) up dA = ∞, a contradiction.

3.4 Minimal surfaces with finite total curvatures In this section, we study the Gauss map of minimal surfaces of finite curvature. See [34, 44]. In particular, we prove Osserman’s result that the Gauss map of a nonflat complete minimal surface in ℝ3 with finite total curvature can omit at most three values. We begin with the following result. Theorem 3.4.1. Let M be a complete Riemanian manifold of dimension 2 whose Gauss curvature K ≤ 0. Assume that ∫ |K|dA < ∞. M

Then there exist a compact 2-manifold M,̃ a finite number of points p1 , . . . , pk on M,̃ and an isometry between M and M̃ − {p1 , . . . , pk }. This is a deep result, and its proof is not presented in this book. See Theorem 9.1 in Osserman’s book [62] for the proof. As we have seen earlier, the Gauss curvature of minimal surfaces is always nonpositive, so complete minimal surfaces of ℝ3 with finite total curvature satisfy the hypothesis of the above theorem. We claim that after identification of M with M̃ − {p1 , . . . , pk }, the Gauss map g : M → ℙ1 (ℂ), which is holomorphic, extends to a meromorphic function on M.̃ Indeed, if any of the points pj were an essential singularity of g, then by the big Picard theorem in complex analysis, g would assume all values of ℙ1 (ℂ) infinitely many times, with at most two exceptions. Since the Gauss map g is the unit normal, this means that the unit normal n would assume all values of S 2 (1) (the unit sphere) infinitely many times, with at most two exceptions. This would imply that the total curvature of M would be infinite (by the geometric meaning of the Gauss curvature; see Theorem 1.2.3), which is contrary to our assumption. Hence g has at most a pole at pj for 1 ≤ j ≤ k. Thus it can be extended to a meromorphic function on M.̃ This proves the claim. Recall that under the isothermal coordinates (u, v) and the immersion x : M → ℝ3 , , where z = u + iv, then αk = ϕi dz, 1 ≤ k ≤ 3, are global holomorphic 1-forms if ϕ := 𝜕x 𝜕z on M, where ϕ = (ϕ1 , ϕ2 , ϕ3 ). Also recall that the induced metric is ds2 = |f |2 (1+|g|2 )2 |dz|2 , where f = ϕ1 − iϕ2 . We have that the form ω := fdz extends as a meromorphic form to M.̃ Indeed, by changing coordinates in ℝ3 we may assume that g has no poles at the points p1 , . . . , pk . Since the metric of M is complete, we have 󵄨 󵄨 lim 󵄨󵄨 f (z)󵄨󵄨󵄨 = ∞.

z→pj 󵄨

3.4 Minimal surfaces with finite total curvatures � 71

Hence f extends as a meromorphic with at most poles at pj , 1 ≤ j ≤ k, on M.̃ Since ϕj , 1 ≤ j ≤ 3, can be expressed through f and g, we see that αk , 1 ≤ k ≤ 3, extend to meromorphic forms on M.̃ Proposition 3.4.2. Let M be a complete minimal surfaces in ℝ3 with finite total curvature. Then ∫ |K|dA = 4π deg(g), M

where g : M̃ → ℙ1 (ℂ) is the holomorphic extension of g : M → ℙ1 (ℂ). Proof. The induced metric is ds2 = 2‖ϕ‖2 |dz|2 = |f |2 (1 + |g|2 )2 |dz|2 , where ϕ = ϕ1 , ϕ2 , ϕ2 .

Its associated metric form is ψ := ‖ϕ‖2 Ric(ψ) = −Kψ and (2.12), we have

√−1 dz ∧ d z̄ 2π

= |f |2 (1 + |g|2 )2

√−1 dz ∧ d z.̄ Hence, using 4π

∫ |K|dA = − ∫ KdA = − ∫ K2‖ϕ‖2 dudv = −2 ∫ K‖ϕ‖2 du ∧ dv M

M̃

M̃

= −2 ∫ K‖ϕ‖2 M̃

M̃

√−1 dz ∧ d z̄ = −2π ∫ Kψ = 2π ∫ Ric(ψ) 2 M̃

c

M

2 2

2

= 2π ∫ dd log(| f | (1 + |g| ) ) M̃ 2

= 2π ∫ dd c log(1 + |g|2 ) = 4π deg(g).

(3.22)

M̃

The induced metric ds2 = 2 ∑3i=1 |ϕi (z)|2 |dz|2 extends to a pseudo-metric on M.̃ Write ds = |z|2νj h2 |dz|2 locally near pj with h(0) ≠ 0 and a local coordinate z with z(pj ) = 0. We prove that 2

νj ≤ −2.

(3.23)

To do so, we prove that the form α = (α1 , α2 , α3 ), where αj = ϕj dz, has a pole of order mj ≥ 2 at each pj . Indeed, take a local coordinate z around pj with z(pj ) = 0. At z = 0, αi has at most a pole of order mij at pj . Since M̃ − {p1 , . . . , pk } is complete in the induced metric given by ds2 = ∑3i=1 |ϕi (z)|2 |dz|2 near pj , we have 3

󵄨 󵄨2 lim ∑󵄨󵄨󵄨ϕi (z)󵄨󵄨󵄨 = ∞.

z→pj

i=1

Thus mj = max{m1j , m2j , m3j } ≥ 1. Observe that mj is exactly the order of the pole of α at pj . Assume that m1 = 1. Then

72 � 3 Minimal surfaces in ℝ3

ϕi (z) =

i ∞ c−1 + ∑ cni zn , z n=0

i = 1, 2, 3.

Since cni n+1 z ) + xi (z0 ), n+1 n=0 ∞

xi = Re ∫ ϕi (z)dz = Re(ci log z + ∑

i we know that the term Re(c−1 log z) must be single valued because all other terms are single valued, which implies that ci must be real. On the other hand, 3

0 = ∑ ϕ2i = i=1

1 2 2 2 3 2 (c−1 ) + (c−1 ) + (c−1 ) d−1 ∞ + + ∑ dn zn , z z2 n=0

i 2 and it follows that ∑3i=1 (c−1 ) = 0, so ci = 0 for i = 1, 2, 3. However, this is impossible since M is complete with respect to ds2 . Therefore we have that αj = ϕj dz has a pole of order mj ≥ 2 at each pj . From the definition of ds2 we see that (3.23) holds. By applying Theorem 2.4.1 (Gauss–Bonnet for pseudo-metric) to ψ = |f |2 (1 + √ |g|2 )2 4π−1 dz ∧ d z,̄ the pseudo-metric form associated with ds2 , we get k

̃ + ∑ νj . − ∫ Ric(ψ) = χ(M) j=1

M̃

By (3.22), 2

2

− ∫ Ric(ψ) = − ∫ dd c log(| f |2 (1 + |g|2 ) ) = − ∫ dd c log(1 + |g|2 ) = −2 deg(g). M̃

M̃

M̃

On the other hand, from (3.23) we get νj ≤ −2 for j = 1, 2, . . . , k. Hence k

̃ + ∑ νj ≤ χ(M) ̃ − 2k. − 2 deg(g) = χ(M) j=1

(3.24)

Theorem 3.4.3 ([61]). Let x : M → ℝ3 be a nonflat complete minimal surface of finite total curvature. Then its Gauss map can omit at most three points (i. e., three directions on the unit sphere). Proof. Since the minimal surface x : M → ℝ3 has finite total curvature, as we discussed above, M is conformally equivalent to a compact surface M̃ punctured at a finite number of points P = p1 , . . . , pk , and the Gauss map g can be extended to a meromorphic function on M.̃ Hence g : M̃ → ℙ1 (ℂ) is a holomorphic map. Assume that g omits a1 , . . . , aq ∈ ℙ1 (ℂ). By Theorem 2.4.3 with E = f −1 {a1 , . . . , aq } we have

3.4 Minimal surfaces with finite total curvatures

� 73

(q − 2)d ≤ |E| + 2(gM̃ − 1) ≤ k + 2(gM̃ − 1), where d = deg(g), and gM̃ is the genus of M.̃ On the other hand, from (3.24), gM̃ −1 ≤ d−k. Therefore (q − 2)d ≤ k + 2(gM̃ − 1) ≤ k + 2(d − k) = 2d − k, or k ≤ (4 − q)d. Since k > 0, we conclude that q < 4.

4 The Nevanlinna theory In this chapter, we introduce the theory of meromorphic functions developed by Nevanlinna [50] in 1925, as well its extension to holomorphic curves in the complex projective space. For references, see [76, 9, 10, 19, 34, 50, 51, 52, 80]. Other related works and recent developments include [8, 15, 17, 21, 23, 36, 37, 69, 70, 74, 75, 78, 81, 82]. The fundamental tool of this subject is the measurement of the growth of f . There are two different ways of measuring an entire function f : its rate of growth—its maximum modulus on the disc of radius r (viewed as a function of r) and the maximum number of times the value in the image is taken on this disc. The insight is that these two rates of growth are essentially the same, the former being roughly the exponential of the latter. For meromorphic functions, the maximum modulus no longer exists if the function has poles. Nevanlinna [50] found the right substitute for the maximum modulus. He introduced the characteristic function Tf (r) to measure the growth of a meromorphic function f . Starting from the Poisson–Jensen formula, he was able to derive a more subtle growth estimate for meromorphic functions, which he called the second main theorem. It gives a quantitative extension of the classical Picard theorem for meromorphic functions. The theory of holomorphic curves, which extends the theory of meromorphic functions, is mainly due to Cartan [10] and Ahlfors [1, 2]. It is the heart part of the so-called Nevanlinna theory. In the mid-1960s, Chern and Osserman [20] initiated a project by using the theory of holomorphic curves to study value distribution properties of the Gauss maps of the complete minimal surfaces in ℝm .

4.1 Nevanlinna theory of meromorphic functions A function f defined on a domain in D ⊂ ℂ is called a meromorphic function on D if it is holomorphic on D except the poles. In this section, we introduce the Nevanlinna theory of meromorphic functions on D(R) = {z | |z| < R} with 0 < R ≤ ∞. The reference book is [40]. The theory of meromorphic functions gives a quantitative formulation of the little Picard theorem. The main tool is the integration result of the so-called logarithmic derivative lemma. We begin with the following Poisson–Jensen formula. Theorem 4.1.1 (Poisson–Jensen formula). Let f ≢ 0 be a meromorphic function on the closed disc D(R), where D(R) = {z | |z| < R}, R < ∞. Let a1 , . . . , ap denote the zeros of f in D(R), counting multiplicities, and let b1 , . . . , bq denote the poles of f in D(R), also counting multiplicities. Then, for any z in |z| < R that is not a zero or pole, we have 2π

R2 − |z|2 󵄨 󵄨 󵄨 󵄨 dθ log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = ∫ log󵄨󵄨󵄨 f (Reiθ )󵄨󵄨󵄨 iθ 2 2π | Re −z| 0

https://doi.org/10.1515/9783110989557-004

4.1 Nevanlinna theory of meromorphic functions

� 75

p 󵄨󵄨 R2 − b̄ j z 󵄨󵄨 󵄨󵄨 R2 − ā z 󵄨󵄨 q 󵄨 󵄨 󵄨󵄨 i 󵄨󵄨 − ∑ log󵄨󵄨󵄨 󵄨󵄨 + ∑ log󵄨󵄨󵄨 󵄨. 󵄨󵄨 R(z − bj ) 󵄨󵄨󵄨 󵄨󵄨 R(z − ai ) 󵄨󵄨 j=1 i=1

Proof. Note that it suffices to prove the theorem when f has no zeros or poles on the circle |z| = R. Otherwise, we consider the function f (ρz) and let ρ → 1. We first consider the case where f is analytic and has no zeros in the closed disc |z| ≤ R. In this case, log |f | is harmonic. For a given z in D(R), we consider the linear 2 transformation L(w) = RR2(z−w) . The function L sends z to zero and satisfies |L(w)| = R ̄ −zw if |w| = R. Let F(w) = log |f (L(w))|. Applying Theorem 2.2.1 (mean value theorem) to harmonic functions to F(w), we have 2π

dw dθ 󵄨 󵄨 = ∫ F(w) . log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = F(0) = ∫ F(Reiθ ) 2π 2πiw 0

(4.1)

|w|=R

Let ζ = L(w). Then w = L−1 (ζ ) =

R2 (z − ζ ) . R2 − zζ̄

So, for |ζ | = R, dw −1 1 −1 z̄ z̄ dζ )dζ = ( = ( + 2 + ) ̄ 2πiw 2πi z − ζ R − zζ̄ z − ζ ζ ζ − zζ̄ 2πi 2 ̄ z dζ R − |z|2 dζ −ζ + ) = . =( z − ζ ζ ̄ − z̄ 2πiζ |ζ − z|2 2πiζ dζ iζ

Note that when |w| = R, |ζ | = R, and

(4.2)

= dθ, so by combining (4.1) and (4.2)

2π

2 2 󵄨 󵄨 󵄨 󵄨 R − |z| dθ log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = ∫ log󵄨󵄨󵄨 f (Reiθ )󵄨󵄨󵄨 . | Reiθ −z|2 2π 0

Thus 2π

2 2 󵄨 󵄨 󵄨 󵄨 R − |z| dθ log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = ∫ log󵄨󵄨󵄨 f (Reiθ )󵄨󵄨󵄨 . iθ | Re −z|2 2π 0

Thus the theorem holds in this case. For the general case, we consider the function g(z) = f (z)

p

R2 −ā μ z R(z−aμ )

q

R2 −b̄ν z R(z−bν )

∏μ=1 ∏ν=1

.

(4.3)

76 � 4 The Nevanlinna theory Then g has no zeros or poles in |z| ≤ R. Note that when |z| = R, |g(z)| = |f (z)|. Applying (4.3) to g yields the theorem. Applying the above theorem with f (z) ≡ e, we have the following: Corollary 4.1.2. Let |z| < R. Then 2π

∫ 0

1 dθ 1 = . | Reiθ −z|2 2π R2 − |z|2

We now define Nevanlinna functions. Let f be a meromorphic function on D(R0 ), where 0 < R0 ≤ ∞, and let r < R0 . Denote the number of poles of f on the open disc D(r) by nf (r, ∞), counting multiplicities. Define the counting function r

Nf (r, ∞) = nf (0, ∞) log r + ∫[nf (t, ∞) − nf (0, ∞)] 0

dt , t

where nf (0, ∞) is the multiplicity if f has a pole at z = 0. For each complex number a, we define Nf (r, a) = N1/(f −a) (r, ∞).

(4.4)

We note that Nf (r, a) measures how many times f takes the value a in |z| < r. By the definition of the Lebesgue–Stieltjes integral it follows that Nf (r, 0) = (ord+0 f ) log r +

󵄨󵄨 r 󵄨󵄨 󵄨 󵄨 ∑ (ord+z f ) log󵄨󵄨󵄨 󵄨󵄨󵄨, 󵄨󵄨 z 󵄨󵄨 |z| 0 and 0 < p < 1. For every a ∈ ℂ, we have 2π

∫ 0

(2 − p) dθ rp ≤ . iθ p |re − a| 2π 2(1 − p)

Proof. Without no loss of generality, we may assume that a is a positive real number. Then for |θ| ≤ π/2, we have 2 󵄨󵄨 iθ 󵄨 󵄨󵄨re − a󵄨󵄨󵄨 ≥ r| sin θ| ≥ r|θ|, π and for π/2 < |θ| ≤ π, we have |reiθ − a| ≥ r. Therefore π/2

2π

π

p

rp π dθ ≤ 2 ∫ ( ) dθ + 2 ∫ dθ ∫ iθ 2θ |re − a|p 0

0

1−p p

≤

2 π π ( ) 1−p 2

1−p

π/2

+π =

π(2 − p) . 1−p

Proposition 4.1.6. Let f be a non-constant meromorphic function in D(R0 ) (0 < R0 ≤ ∞). Let p and p′ be real numbers such that 0 < p < p′ < 1. Let 0 < r0 < R0 . Then there exists a positive constant K such that for r0 < r < ρ < R0 , 2π

p 󵄨󵄨 f ′ 󵄨󵄨p dθ ρ 󵄨 󵄨 ≤ K( Tf (ρ)) . ∫ 󵄨󵄨󵄨r( )(reiθ )󵄨󵄨󵄨 󵄨󵄨 f 󵄨󵄨 2π ρ−r 0

′

(4.9)

4.1 Nevanlinna theory of meromorphic functions

� 79

Proof. We start from the Poisson–Jensen formula: for 0 < r < s < R0 , 2π

󵄨󵄨 s2 − ā μ z 󵄨󵄨 s2 − |z|2 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 iϕ 󵄨󵄨 dϕ f (se ) log󵄨󵄨󵄨 f (z)󵄨󵄨󵄨 = ∫ iϕ log − log ∑ 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 2π 󵄨 󵄨󵄨 s(z − aμ ) 󵄨󵄨󵄨 |se − z|2 |a | 0, we have the inequality q

(q − (n + 1))Tf (r) ≤ ∑ Nf(n) (r, Hj ) + S(r) + j=1

n(n + 1) ϵ log+ r 2

98 � 4 The Nevanlinna theory for 0 < r < R0 , where S(r) is evaluated as follows: (1) In the case R0 < ∞, S(r) ≤ C(log+

1 + log+ Tf (r)) R0 − r

for every r ∈ [0, R0 ) excluding a set E such that ∫E (2) In the case R0 = ∞, for every ϵ > 0, S(r) ≤

1 dt R0 −t

< ∞.

n(n + 1) ((1 + ϵ) log+ Tf (r) + (1 + ϵ)2 log+ log Tf (r)) + O(1) 2

for every r ∈ [0, +∞) excluding a set E ′ such that ∫E′ dt < ∞. Note that Theorem 4.2.2 assumes that H1 , . . . , Hq are in general position. Now we consider the case that H1 , . . . , Hq are not necessarily in general position. Let K ⊂ {1, . . . , q} be a set such that ak , k ∈ K, are linearly independent. Assuming that #K = n + 1, it is easy to prove the following: there is a constant C1 such that 󵄨󵄨 W (f , . . . , f ) 󵄨󵄨 󵄨󵄨 W (Fi , . . . , Fi ) 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−n−1 1 n+1 󵄨󵄨 0 n 󵄨󵄨 ≤ C1 ( ∑ 󵄨󵄨), 󵄨󵄨‖f ‖ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∏j∈K ⟨f, aj ⟩ 󵄨󵄨 󵄨󵄨 Fi ⋅ ⋅ ⋅ Fi 1 n+1 1≤i1 0 2

and for l = 2n +4n en d 2n (nI(ϵ−1 ))n , where d is the least common multiple of the dj , we have q

(q − (n + 1) − ϵ)Tf (r) ≤ ∑ dj−1 Nf(l−1) (r, Dj ), j=1

where the inequality holds for all r ∈ (0, +∞) except for a possible set E of finite Lebesgue measure. Here I(x) := min{a ∈ ℤ | a > x} for a real number x. Define the defect of f for a hypersurface D of degree d: f ,⋆ δl−1 (D)

= 1 − lim sup r→+∞

Nf(l−1) (r, D) dTf (r)

.

Then Theorem 4.2.7 gives the following defect relation. Corollary 4.2.8 (Defect relation). Let f : ℂ → ℙn (ℂ) be an algebraically nondegenerate holomorphic map, and let D1 , . . . , Dq be hypersurfaces in ℙn (ℂ) in general position. Then we have q

f ,⋆

∑ δl−1 (Dj ) ≤ n + 1. j=1

Before we prove Theorem 4.2.7, we first prove some preparation lemmas.

100 � 4 The Nevanlinna theory Lemma 4.2.9. Let ϕ1 , . . . , ϕm be homogeneous polynomials in ℂ[X0 , . . . , Xn ]. Assume that they define a subvariety of ℙn (ℂ) of dimension n − m. Then {ϕ1 , . . . , ϕm } is a regular sequence, that is, for i = 1, . . . , m, the element ϕi is not a divisor of the zero in the ring ℂ[X0 ,...,Xn ] . (ϕ ,...,ϕ ) 1

i−1

Proof. This is a well-known fact from the theory of Cohen–Macaulay rings. We will use the lexicographic ordering on the m-tuples (i1 , . . . , im ) ∈ ℕm of natural numbers. Namely, (j1 , . . . , jm ) > (i1 , . . . , im ) iff for some b ∈ {1, . . . , m}, we have jl = il for l < b and jb > ib . Lemma 4.2.10. Let R be a commutative ring, and let ϕ1 , . . . , ϕm be a regular sequence in R. Denote by I ⊂ R the ideal generated by ϕ1 , . . . , ϕm . Suppose that for some q, q1 , . . . , qh ∈ R, we have the equation i

h

j (r)

ϕ11 ⋅ ⋅ ⋅ ϕimm q = ∑ ϕ11 r=1

⋅ ⋅ ⋅ ϕjmm (r) qr ,

where (j1 (r), . . . , jm (r)) > (i1 , . . . , im ) (in the lexicographic order) for r = 1, . . . , h. Then q ∈ I. Proof. We argue by induction on m. Since ϕ1 is not a zero divisor in R, the statement is trivial for m = 1. Assume that m > 1 and the lemma is true up to m − 1. Renumbering the indices 1, . . . , h, we may assume that j1 (r) > i1 for r = 1, . . . , s (possibly, s = 0 or s = h) and that j1 (r) = i1 for r = s + 1, . . . , h (the case j1 (r) < i1 is excluded since (j1 (r), . . . , jm (r)) > (i1 , . . . , im )). Since ϕ1 is not a zero divisor in R, we may then write h

i

j (r)

ϕ22 ⋅ ⋅ ⋅ ϕimm q = ϕ1 σ + ∑ ϕ22 r=s+1

⋅ ⋅ ⋅ ϕjmm (r) qr ,

where σ ∈ R. We now reduce modulo ϕ1 , denoting the reduction with a bar, and work in the ring R′ = R/(ϕ1 ). We obtain h

i j (r) ϕ̄ 22 ⋅ ⋅ ⋅ ϕ̄ imm q̄ = ∑ ϕ̄ 22 ⋅ ⋅ ⋅ ϕ̄ jmm (r) q̄ r . r=s+1

Note that (j2 (r), . . . , jm (r)) > (i2 , . . . , im ) for r = s + 1, . . . , m and that {ϕ̄ 2 , . . . , ϕ̄ m } is a regular sequence in R′ . We may thus apply the inductive assumption with m − 1 in place of m and R′ in place of R. We obtain that q̄ lies in the ideal of R′ generated by ϕ̄ 2 , . . . , ϕ̄ m , that is, q ∈ I, as required. This finishes the proof. Lemma 4.2.11. Let VN be the space of homogeneous polynomials of n + 1 variables of degree N. Let Q1 , . . . , Qq , q ≥ n + 1, be a set of homogeneous polynomials of n + 1 variables of the same degree d ≥ 1 and located in general position. Then for any nonnegative integer VN N and any {j1 , . . . , jn } ⊂ {1, . . . , q}, the dimension of the vector space (Q ,...,Q is equal to )∩V j1

jn

N

4.2 Cartan’s second main theorem for holomorphic curves

� 101

the number of n-tuples (i) = (i1 , . . . , in ) ∈ ℤn≥0 such that i1 + ⋅ ⋅ ⋅ + in ≤ N and 0 ≤ i1 , . . . , in ≤ d − 1. In particular, for all N ≥ n(d − 1), we have dim

VN = dn . (Qj1 , . . . , Qjn ) ∩ VN

Proof. The case N = 0 holds trivially, so we assume that N is positive for the rest of the proof. We first prove that dim

VN VN = dim (Qj1 , . . . , Qjn ) ∩ VN (Q1 , . . . , Qn ) ∩ VN

(4.38)

for any choice of {j1 , . . . , jn } ⊂ {1, . . . , q} and any N. For this, it suffices to prove that dim((Q1 , . . . , Qn ) ∩ VN ) = dim((Qj1 , . . . , Qjn ) ∩ VN ). Since the order of the Qj does not matter, we only need to prove that dim((Q1 , . . . , Qn ) ∩ VN ) = dim((Q1 , . . . , Qn−1 , Qjn ) ∩ VN ),

(4.39)

and the rest follows by induction. Notice that to prove (4.39), we only need to show that dim

(Q1 , . . . , Qn−1 , Qjn ) ∩ VN (Q1 , . . . , Qn ) ∩ VN = dim . (Q1 , . . . , Qn−1 ) ∩ VN (Q1 , . . . , Qn−1 ) ∩ VN

(4.40)

To do so, consider the map ϕ:

(Q1 , . . . , Qn−1 , Qjn ) ∩ VN (Q1 , . . . , Qn ) ∩ VN → (Q1 , . . . , Qn−1 ) ∩ VN (Q1 , . . . , Qn−1 ) ∩ VN n−1

n−1

j=1

j=1

[ ∑ bj Qj + bn Qn ] 󳨃→ [ ∑ bj Qj + bn Qjn ] with bj ∈ ℂ[x0 , . . . , xn ]. This map is clearly surjective, so we only need to prove that it is well defined and injective. To prove that ϕ is well defined, let [∑n−1 j=1 bj Qj + bn Qn ] = n−1 [∑j=1 bj Qj + b′n Qn ]. This means that (bn − b′n )Qn ∈ (Q1 , . . . , Qn−1 ) ∩ VN . Since Q1 , . . . , Qn is

a regular sequence by Lemma 4.2.9, Qn is not a zero divisor in n−1

n−1

j=1

j=1

ℂ[x0 ,...,xn ] . (Q1 ,...,Qn−1 )

Hence

[ ∑ bj Qj + bn Qjn ] − [ ∑ bj Qj + b′n Qjn ] = [(bn − b′n )Qjn ] = 0 in

(Q1 ,...,Qn−1 ,Qjn )∩VN (Q1 ,...,Qn−1 )∩VN

. So ϕ is well defined. The injectivity of ϕ follows by the same argument,

just changing the roles of Qn and Qjn , and using the fact that Q1 , . . . , Qn−1 , Qjn is also a regular sequence by Lemma 4.2.9. Hence we proved that (4.40) is true, and thus (4.38)

102 � 4 The Nevanlinna theory holds. We remark that for the proof, we only used the fact that Q1 , . . . , Qn , Qjn are in general position. Since Q1 , . . . , Qn define a subvariety of dimension 0, there exists a hyperplane Hn+1 on ℙn (ℂ) such that ⋂nj=1 {Qj = 0} ∩ Hn+1 = 0. Furthermore, by induction there exist n hyperplanes H1 , . . . , Hn+1 such that ⋂i−1 j=1 {Qj = 0} ∩ (⋂k=i Hk ) = 0. This means that

Q1 , . . . , Qi−1 , Hid , . . . , Hnd are in general position. Then by (4.39) and by induction we get that dim

VN VN . = dim d (Q1 , . . . , Qn ) ∩ VN (H1 , . . . , Hnd ) ∩ VN

Let L1 , . . . , Ln+1 be the linear forms defining H1 , . . . , Hn+1 . Since L1 , . . . , Ln+1 are linearly independent, it follows from a well-known fact of linear algebra that there exists a permutation {k1 , . . . , kn+1 } of {1, . . . , n + 1} such that {L1 , . . . , Li−1 , xki , . . . , xkn+1 } are linearly independent for any i ∈ {1, . . . , n + 2}. This means that {Ld1 , . . . , Ldi−1 , xkdi , . . . , xkdn+1 } are in general position. Hence by (4.39) and by induction we get that dim

VN d (L1 , . . . , Ldn )

dim

VN VN = dim d . (Qj1 , . . . , Qjn ) ∩ VN (x1 , . . . , xnd ) ∩ VN

∩ VN

= dim

VN d (x1 , . . . , xnd )

∩ VN

,

and thus

On the other hand, it is easy to see that for any positive integer N, the vector space N−(i +⋅⋅⋅+in ) i1 i VN has a basis {x0 1 x1 ⋅ ⋅ ⋅ xnn | i1 + ⋅ ⋅ ⋅ + in ≤ N, 0 ≤ i1 , . . . , in ≤ d − 1}. This (x d ,...,x d )∩V 1

n

N

completes the proof.

Let D1 , . . . , Dq be hypersurfaces in ℙn (ℂ) located in general position. Let Qj , 1 ≤ j ≤ q, be the homogeneous polynomials in ℂ[X0 , . . . , Xn ] of degree dj defining Dj . Replacing Qj d/d

by Qj j if necessary, where d is the l. c. m. of the dj , we can assume that Q1 , . . . , Qq have the same degree d. For N ∈ ℕ, let VN be the space of homogeneous polynomials of degree N of n + 1 variables. Pick n distinct polynomials γ1 , . . . , γn ∈ {Q1 , . . . , Qq }. Arrange the n-tuples i = (i1 , . . . , in ) of nonnegative integers by lexicographic order. For the n-tuples i = (i1 , . . . , in ) of nonnegative integers with σ(i) := ∑j ij ≤ N/d, define the spaces e

Wi = WN,i := ∑ γ1 1 ⋅ ⋅ ⋅ γnen VN−dσ(e) . e≥i

(4.41)

Clearly, W(0,...,0) = VN and Wi ⊃ Wi′ if i′ ≥ i, so that {Wi } in fact defines a filtration of VN . Suppose that i′ follows i in the ordering. We have the following lemma.

� 103

4.2 Cartan’s second main theorem for holomorphic curves

Lemma 4.2.12. There is an isomorphism VN−dσ(i) Wi ≅ . Wi′ (γ1 , . . . , γn ) ∩ VN−dσ(i) Proof. Define the vector space homomorphism ϕ : VN−dσ(i) →

Wi Wi′

as follows: for a i

i

polynomial q(X0 , . . . , Xn ) in VN−dσ(i) , we define ϕ(q) as the class of γ11 ⋅ ⋅ ⋅ γnn q (which belongs to Wi ) modulo Wi′ . By our definition of the spaces We this homomorphism is i i surjective. To find its kernel, suppose that q ∈ ker ϕ. This means that γ11 ⋅ ⋅ ⋅ γnn q lies in en e1 ∑(e)>(i) γ1 ⋅ ⋅ ⋅ γn VN−dσ(e) . Then we may write i

e

γ11 ⋅ ⋅ ⋅ γnin q = ∑ γ1 1 ⋅ ⋅ ⋅ γnen q(e) (e)>(i)

for elements q(e) ∈ VN−dσ(e) . By Lemma 4.2.10, q lies in the ideal generated by γ1 , . . . γn . Therefore q = ∑nj=1 αj γj , where αj , 1 ≤ j ≤ n, are homogeneous with deg αj = i

i

deg q − d. Then αj ∈ VN−d(σ(i)+1) . Thus we see that γ11 ⋅ ⋅ ⋅ γnn q is a sum of terms in Wi′ , which concludes the proof of the lemma.

Let △i = dim(Wi /Wi′ ), where i′ > i are consecutive n-tuples with Wi′ ⊂ Wi . By Lemma 4.2.11, △i = d n for every i such that N − dσ(i) ≥ n(d − 1). Moreover, Lemma 4.2.11 implies that △i is independent of the choice of γ1 , . . . , γn . Hence ∑i ij △i is independent of the choice of γ1 , . . . , γn and of j = 1, . . . , n. Ffor 1 ≤ j ≤ n, set △ := ∑ ij △i .

(4.42)

i

Lemma 4.2.13. With N = 2d(n + 1)(nd + n)(2n − 1)(I(ϵ−1 ) + 1) + nd for any ϵ > 0, we have lN ≤ d(n + 1) + ϵ/2, △ where l = ( N+n n ). Moreover, l satisfies the estimate 2

n

l ≤ 2n +4n en d 2n (nI(ϵ−1 )) , where I(x) := min{k ∈ ℕ : k > x} for a positive real number x. Proof. First, notice that l=(

N +n (n + N)(n + N − 1) ⋅ ⋅ ⋅ (N + 1)N! (N + n)n )= ≤ . n N!n! n!

Now since N is divisible by d, it follows from Lemma 4.2.11 that

104 � 4 The Nevanlinna theory

△=

∑

σ(i)≤N/d

ij △i ≥

∑

σ(i)≤N/d−n

ij △i = d n

∑

ij

dn n+1

̂ σ(i)=N/d−n

σ(i)≤N/d−n

n+1

=

dn n+1

=

N/d dn N(N − d) ⋅ ⋅ ⋅ (N − dn) ( )(N/d − n) = , n n+1 d(n + 1)!

∑

∑ ij =

̂ j=1 σ(i)=N/d−n

∑

(N/d − n)

where i ̂ = (i1 , . . . , in+1 ), and we used the fact that the number of nonnegative integer m-tuples with sum ≤ T for a positive integer T is equal to the number of nonnegative T +m integer (m + 1)-tuples with sum exactly T, that is, to ( ). m Since (N − dj) ≥ (N − dn) for every integer j ≤ n, we have n

∏ j=1

n

1 1 ≤( ) , N − dj N − dn

and thus n

N +n lN ≤ d(n + 1)( ) . △ N − nd Using N = 2d(n + 1)(nd + n)(2n − 1)(I(ϵ−1 ) + 1) + nd, we find that n

(

n

r

n n N +n n + nd nd + n ) = (1 + ) = 1 + ∑( )( ) r N − nd N − nd N − nd r=1

≤ 1 + (2n − 1)

nd + n ϵ ≤1+ . N − nd 2d(n + 1)

Therefore lN ≤ d(n + 1) + ϵ/2. △ To estimate l, we use the inequality x

(

y

x+y (x + y)x+y y x x )≤ = (1 + ) (1 + ) ≤ (e(1 + )) y x x yy x y y

y

for positive integers x, y. Hence, with N = 2d(n + 1)(nd + n)(2n − 1)(I(ϵ−1 ) + 1) + nd, we have

4.2 Cartan’s second main theorem for holomorphic curves

� 105

n

N +n N ) ≤ en (1 + ) n n

l=(

≤ en (1 + 2d(n + 1)(d + 1)(2n − 1)(I(ϵ−1 ) + 1) + d)

n

n

2

≤ 2n +4n en d 2n (nI(ϵ−1 )) . This proves the lemma.

Proof of Theorem 4.2.7. Let Qj , 1 ≤ j ≤ q, be the homogeneous polynomials in ℂ[X0 , . . . , d/d

Xn ] of degree dj defining Dj . Replacing Qj by Qj j if necessary, where d is the l. c. m. of dj , we can assume that Q1 , . . . , Qq have the same degree of d. Let N ∈ ℕ, and let VN be the space of homogeneous polynomials of n + 1 variables of degree N and fix an (arbitrary) basis ϕ1 , . . . , ϕl , where l = dim VN . Set F = [ϕ1 (f ) : ⋅ ⋅ ⋅ : ϕl (f )]. Since f is algebraically nondegenerate, F : ℂ → ℙl−1 (ℂ) is linearly nondegenerate. Given z ∈ ℂ, there exists a numbering {i1 , . . . , iq } of the indices 1, . . . , q such that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(Qi1 ∘ f )(z)󵄨󵄨󵄨 ≤ ⋅ ⋅ ⋅ ≤ 󵄨󵄨󵄨(Qiq ∘ f )(z)󵄨󵄨󵄨. Since Q1 , . . . , Qq are in general position, Hilbert Nullstellensatz implies that for any integer k, 0 ≤ k ≤ n, there is an integer mk ≥ d such that m

n+1

xk k = ∑ bjk (x0 , . . . , xn )Qij (x0 , . . . , xn ), j=1

where bjk , 1 ≤ j ≤ n + 1, 0 ≤ k ≤ n, are homogeneous forms with coefficients in ℂ of degree mk − d. So 󵄨󵄨 󵄨mk 󵄨 󵄨 󵄨 󵄨 m −d 󵄨󵄨 fk (z)󵄨󵄨󵄨 ≤ c1 ‖f (z)‖ k max{󵄨󵄨󵄨Qi1 (f )(z)󵄨󵄨󵄨, . . . , 󵄨󵄨󵄨Qin+1 (f )(z)󵄨󵄨󵄨}, where c1 is a positive constant depending only on the coefficients of bjk and thus depends only on the coefficients of Qj . Therefore 󵄩󵄩 󵄩d 󵄨 󵄨 󵄨 󵄨 󵄩󵄩f (z)󵄩󵄩󵄩 ≤ c1 max{󵄨󵄨󵄨Qi1 (f )(z)󵄨󵄨󵄨, . . . , 󵄨󵄨󵄨Qin+1 (f )(z)󵄨󵄨󵄨}. Therefore we derive that q

∏ j=1

n ‖f (z)‖d ‖f (z)‖d q−n ≤ c1 ∏ . |Qj (f )(z)| |Qik (f )(z)| k=1

(4.43)

Take γ1 = Qi1 , . . . , γn = Qin and let WN,i be defined by (4.41). Then we get a filtration VN = W0 ⊃ ⋅ ⋅ ⋅ ⊃ Wi ⊃ Wi′ ⊃ ⋅ ⋅ ⋅, where we just write Wi for WN,i . We now choose a basis ψ1 , . . . , ψl for VN in the following way: We start with the last nonzero Wi and pick a basis of it; Then we continue inductively as follows: suppose i′ > i are consecutive n-tuples such that dσ(i), dσ(i′ ) ≤ N and assume that we have chosen a basis of W(i′ ) ;

106 � 4 The Nevanlinna theory It follows directly from the definition that we may pick representatives in Wi for the i i quotient space Wi /Wi′ , of the form γ11 ⋅ ⋅ ⋅ γnn η, where η ∈ VN−dσ(i) . We extend the previously constructed basis in Wi′ by adding these representatives. In particular, we have obtained a basis for Wi , and our induction procedure may go on unless Wi = VN . Note that if we let ψ be an element of the basis constructed with respect to Wi /Wi′ , then we i i may write ψ = γ11 ⋅ ⋅ ⋅ γnn η, where η ∈ VN−dσ(i) . Thus we have the bound 󵄩N−dσ(i) 󵄨in 󵄩 󵄨i1 󵄨 󵄨 󵄨 󵄨󵄨 , 󵄨󵄨ψ(f )(z)󵄨󵄨󵄨 ≤ c2 󵄨󵄨󵄨γ1 (f )(z)󵄨󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨󵄨γn (f )(z)󵄨󵄨󵄨 󵄩󵄩󵄩f (z)󵄩󵄩󵄩 where c2 is a positive constant depending only on f and Q1 , . . . , Qq . Observe that there are precisely △i such functions ψ in our basis. Write ψ1 , . . . , ψl as linear forms L1 , . . . , Ll in ϕ1 , . . . , ϕl such that ψt (f ) = Lt (F), where F = [ϕ1 (f ) : ⋅ ⋅ ⋅ : ϕl (f )]. Then from the above we have l

󵄨 󵄨 󵄨 i 󵄨△ 󵄩 󵄩lN−d ∑i σ(i)△i , ∏󵄨󵄨󵄨Lt (F(z))󵄨󵄨󵄨 ≤ K( ∏ 󵄨󵄨󵄨γ11 (f (z)) ⋅ ⋅ ⋅ γnin (f (z))󵄨󵄨󵄨 i )󵄩󵄩󵄩f (z)󵄩󵄩󵄩 t=1

i=(i1 ,...,in )

where, as we noted earlier, K is a constant depending only on f and D1 , . . . , Dq and may be different each time. So ‖f (z)‖d ∑i σ(i)△i

≤K

i △

i△

∏i |γ11 i (f (z))| ⋅ ⋅ ⋅ |γnn i (f (z))|

‖f (z)‖lN

∏lt=1 |Lt (F(z))|

.

Thus using the definition that △ := ∑i ij △i for each 1 ≤ j ≤ n, we get ‖f (z)‖dn△ ‖f (z)‖lN . ≤ K |γ1△ (f (z))| ⋅ ⋅ ⋅ |γn△ (f (z))| ∏lt=1 |Lt (F(z))| With γ1 = Qi1 , . . . , γn = Qin , it gives ‖f (z)‖dn△ ‖f (z)‖lN . ≤ K |Qi△ (f (z)) ⋅ ⋅ ⋅ Qi△ (f (z))| ∏lt=1 |Lt (F(z))| 1

n

On the other hand, from (4.43) we have ‖f (z)‖dq△ ‖f (z)‖dn△ ≤ K . |Q1△ (f (z)) ⋅ ⋅ ⋅ Qq△ (f (z))| |Qi△ (f )(z) ⋅ ⋅ ⋅ Qi△ (f )(z)| k

n

Hence, also using the fact that c1 ‖f (z)‖N ≤ ‖F(z)‖ ≤ c2 ‖f (z)‖N , we derived the following key estimate: ‖f (z)‖dq△

|Q1△ (f (z)) ⋅ ⋅ ⋅ Qq△ (f (z))|

≤C

‖F(z)‖l

∏lt=1 |Lt (F(z))|

,

(4.44)

4.2 Cartan’s second main theorem for holomorphic curves

� 107

where C is a constant. Note that although L1 , . . . , Ll depend on z, there are only finitely many such choices, and we denote the total collection as {L1 , . . . , Lu }, since there are only finitely many choices of {γ1 , . . . γn } ⊂ {Q1 , . . . , Qq }. Estimate (4.44) implies that q

2π

j=1

0

△ ∑ mf (r, Dj ) ≤ ∫ max log ∏ K

j∈K

‖F(reiθ )‖‖Lj ‖ dθ |Lj (F)(reiθ )| 2π

+ O(1),

where maxK is taken over all subsets K of {1, . . . , u} such that the linear forms Lj , j ∈ K, are linearly independent. Applying Theorem 4.2.6 (with R0 = ∞) with ϵ/(2N) to the holomorphic map F and linear forms L1 , . . . , Lu and using the fact that TF (r) ≤ NTf (r) + O(1), we obtain 2π

∫ max log ∏ 0

K

j∈K

‖F(reiθ )‖‖Lj ‖ dθ |Lj (F)(reiθ )| 2π

≤exc (l + ϵ/(2N))TF (r) − NW (r, 0) ≤ (lN + ϵ/2)Tf (r) − NW (r, 0),

where ≤exc means the inequality for all r outside of a set E of finite Lebesgue measure, and where W = W (F0 , . . . , Fl ). Hence q

△ ∑ mf (r, Dj ) ≤exc (lN + ϵ)Tf (r) − NW (r, 0). j=1

Using the first main theorem this gives q

(qd −

1 lN )T (r) ≤exc ∑ Nf (r, Qj ) − NW (r, 0) + (ϵ/2)Tf (r). △ f △ j=1

From Lemma 4.2.13 with N = 2d(n + 1)(nd + n)(2n − 1)(I(ϵ−1 ) + 1) + nd for any ϵ > 0, we have lN ≤ d(n + 1) + ϵ/2, △ where l = ( N+n n ). Moreover, l satisfies the estimate n

2

l ≤ 2n +4n en d 2n (nI(ϵ−1 )) , where I(x) := min{k ∈ ℕ : k > x} for a positive real number x. With such l, we get q

(d(q − (n + 1))Tf (r) ≤exc ∑ Nf (r, Qj ) − j=1

1 N (r, 0) + ϵTf (r). △ W

108 � 4 The Nevanlinna theory q

We now estimate ∑j=1 Nf (r, Qj )− △1 NW (r, 0). For each z ∈ ℂ, without loss of generality, we assume that Qj (f ) vanishes at z for 1 ≤ j ≤ q1 and Qj (f ) does not vanish at z for j > q1 . By the hypothesis that the Dj are in general position we know that q1 ≤ n. There are integers kj ≥ 0 and nowhere vanishing holomorphic functions gj such that for j = 1, . . . , q, Qj (f ) = (ζ − z)kj gj , where kj = 0 if q1 < j ≤ q. For {Q1 , . . . , Qn } ⊂ {Q1 , . . . , Qq }, as above, we can obtain a basis ψ1 , . . . , ψl of VN and linearly independent linear forms L1 , . . . , Ll such that ψt (f ) = Lt (F), 1 ≤ t ≤ l. From basic properties of Wronskians we have W = W (F1 , . . . , Fl ) = CW (L1 (F), . . . , Ll (F)) = CW (ψ1 (f ), . . . , ψl (f )), where C is some constant. Let ψ be an element of the basis {ψ1 , . . . , ψl }. As we discussed i i earlier, we may write ψ = Q11 ⋅ ⋅ ⋅ Qnn η with η ∈ VN−dσ(i) . Therefore i

i

ψ(f ) = (Q1 (f )) 1 ⋅ ⋅ ⋅ (Qn (f )) n η(f ). i

Note that there are △i such that ψ is our basis. Notice that (Qj (f ))ij = (ζ − z)ij kj gj j for j = 1, . . . , n. Also, we can assume that kj ≥ l − 1 for 1 ≤ j ≤ q0 and 1 ≤ kj < l − 1 for q0 < j ≤ q1 . Then W vanishes at z with order at least q0

∑(∑ ij (kj − (l − 1))) △i i

j=1 q0

q0

= ∑[(∑ ij △i )(kj − (l − 1))] = △ ∑(kj − (l − 1)). j=1

j=1

i

On the other hand, q

n

q0

q1

j=1

j=1

j=1

j=q0

∑ kj = ∑ kj = ∑ kj + ∑ kj . Hence q

∑ Nf (r, Qj ) − j=1

q

1 N (r, 0) ≤ ∑ Nf[l−1] (r, Dj ). △ W j=0

The theorem is thus proved. Finally, we extend the above result by replacing ℙn (ℂ) with an arbitrary complex projective variety X.

4.2 Cartan’s second main theorem for holomorphic curves

� 109

Theorem 4.2.14 ([73]). Let X be a complex projective variety of dimension n ≥ 1. Let D1 , . . . , Dq be hypersurfaces on X located in general position. Assume that each Dj , 1 ≤ j ≤ q, is linearly equivalent to dj A, where A is an ample divisor. Let f : ℂ → X be an algebraically nondegenerate holomorphic map. Then, for every ϵ > 0, q

∑ dj−1 mf (r, Dj ) ≤exc (n + 1 + ϵ)Tf (r), j=1

where ≤exc means the inequality for all r ∈ (0, +∞) except for a possible set E of finite Lebesgue measure. To prove Theorem 4.2.14, we first introduce some notation. Let Y ⊂ ℙM (ℂ) be a projective variety of dimension n (we will take M = q − 1 in the proof). Denote by ℂ[x0 , . . . , xM ]m the vector space of homogeneous polynomials in ℂ[x0 , . . . , xM ] of degree m (including m = 0). Let IY be the prime ideal in ℂ[x0 , . . . , xM ] defining Y . Put (IY )m = ℂ[x0 , . . . , xM ]m ∩ IY . The Hilbert polynomial HY (m) of Y is defined by HY (m) := dim(ℂ[x0 , . . . , xM ]m /(IY )m ). The Hilbert weight of Y with respect to the weight c ∈ ℝM is defined by HY (m)

SY (m, c) = max( ∑ ai ⋅ c), i=1

(4.45)

where the maximum is taken over all sets of monomials xa1 , . . . , xaHY (m) whose residue classes modulo IY form a basis of ℂ[x0 , . . . , xN ]m /(IY )m , where for x = (x0 , . . . , xM ) and a0 aM a a = (a0 , . . . , aM ) ∈ ℤM+1 + , x = x0 ⋅ ⋅ ⋅ xM . We need the following lemma to estimate (get a lower bound) of Hilbert weight SY (m, c). Lemma 4.2.15 ([73]). Let Y ⊂ ℙM be an algebraic variety of dimension n and degree △. Let m > △ be an integer, and let c = (c0 , . . . , cM ) ∈ ℝM+1 . Let {i0 , . . . , in } be a subset of {0, . . . , M} such that {x = [x0 : ⋅ ⋅ ⋅ : xM ] ∈ ℙM | xi0 = ⋅ ⋅ ⋅ = xin = 0} ∩ Y = 0. Then 1 1 (2n + 1)△ S (m, c) ≥ (c + ⋅ ⋅ ⋅ + cin ) − ( max ci ). 0≤i≤M mHY (m) Y (n + 1) i0 m Proof of Theorem 4.2.14. We assume that A is very ample, and we embed X through A into the complex projective space X ⊂ ℙN (ℂ). So we can assume that D1 , . . . , Dq are defined by Q1 , . . . , Qq , where each Qj is a homogeneous polynomial of degree dj in N + d/dj

1 variables. Replacing Qj by Qj

if necessary, where d is the l. c. m. of the dj , we can

110 � 4 The Nevanlinna theory assume that Q1 , . . . , Qq have the same degree of d. Consider the map χ : X → ℙq−1 (ℂ) given by χ(w) = [Q1 (w) : ⋅ ⋅ ⋅ : Qq (w)]. Then it is a finite morphism since D1 , . . . Dq are in general position. Let Y = χ(X). Then Y is a complex subvariety of ℙq−1 (ℂ) with dim Y = dim X = n and △ := deg Y ≤ d n deg X. Let m be a positive integer. Consider the vector space Vm = ℂ[x1 , . . . , xq ]m /(IY )m , and fix a basis ϕ0 , . . . , ϕnm , where nm = HY (m) − 1. Consider the map F = [ϕ0 (χ ∘ f ), . . . , ϕnm (χ ∘ f )] : ℂ → ℙnm . For each z ∈ ℂ, define cj (z) = log

‖f (z)‖d ‖Qj ‖ |Qj (f )(z)|

,

1 ≤ j ≤ q,

and let c(z) = (c1 (z), . . . , cq (z)). With this given weight c(z), take monomials (which depend on z) xa1 , . . . , xaHY (m) whose residue classes modulo IY form a basis of Vm = ℂ[x1 , . . . , xq ]m /(IY )m and which reaches the maximum in the Hilbert weight SY (m, c(z)) (see (4.45)). For each 1 ≤ j ≤ HY (m), write xaj = Lj,z (ϕ0 , . . . , ϕnm ). Then Lj,z are linear forms that are linearly independent for each z. Note that the linear forms depend on z, but there are only finitely many of them in total. We prove the following key estimate: For each z ∈ ℂ and for the linear forms Lj,z , 1 ≤ j ≤ HY (m), chosen above, we have q

∑ log j=1

‖f (z)‖d ‖Qj ‖ |Qj (f )(z)|

H (m)

≤

(n + 1) Y ‖F(z)‖ 󵄩 󵄩 + d(n + 1) log󵄩󵄩󵄩f (z)󵄩󵄩󵄩 ∑ log mHY (m) j=1 |Lj,z (F)(z)| ‖f (z)‖d ‖Qi ‖ n+1 󵄩 󵄩 (n + 1)(2n + 1)△ log󵄩󵄩󵄩F(z)󵄩󵄩󵄩 + (max log ) 0≤i≤q m m |Qi (f )(z)| + O(1). (4.46) −

Indeed, since D1 , . . . , Dq are in general position, for every z ∈ ℂ, there are i0 , . . . , in (which may depend on z) such that q

‖f (z)‖d ‖Qj ‖

j=1

|Qj (f )(z)|

∑ log With cj (z) = log

‖f (z)‖d ‖Qj ‖ |Qj (f )(z)|

n

‖f (z)‖d ‖Qit ‖

t=0

|Qit (f )(z)|

≤ ∑ log

.

and c(z) = (c1 (z), . . . , cq (z)), by applying Lemma 4.2.15 we get

1 1 (2n + 1)△ S (m, c(z)) ≥ (c (z) + ⋅ ⋅ ⋅ + cin (z)) − (max ci (z)), 1≤i≤q mHY (m) Y (n + 1) i0 m that is,

4.2 Cartan’s second main theorem for holomorphic curves

n

∑ log

t=0

‖f ‖d ‖Qit ‖ |Qit (f )(z)|

� 111

≤

‖f (z)‖d ‖Qi ‖ (n + 1)SY (m, c(z)) (n + 1)(2n + 1)△ + (max log ). 0≤i≤q mHY (m) m |Qi (f )(z)|

≤

‖f (z)‖d ‖Qi ‖ (n + 1)SY (m, c(z)) (n + 1)(2n + 1)△ + (max log ). 0≤i≤q mHY (m) m |Qi (f )(z)|

Therefore q

∑ log

‖f (z)‖d ‖Qj ‖

j=1

|Qj (f )(z)|

However, by our selection of the linear forms Lj,z , Lj,z (ϕ0 , . . . , ϕnm ) = xaj , and HY (m)

SY (m, c(z)) = ∑ (ai ⋅ c(z)). i=1

Hence ai1

Li,z (F(z)) = (Q1 (f )(z))

a

⋅ ⋅ ⋅ (Qq (f )(z)) iq ,

where ai = (ai1 , . . . , aiq ). Thus 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 − log󵄨󵄨󵄨Li,z (F(z))󵄨󵄨󵄨 = −(ai1 log󵄨󵄨󵄨Q1 (f )(z)󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + aiq log󵄨󵄨󵄨Qq (f )(z)󵄨󵄨󵄨) 󵄩 󵄩d = ai1 c1 (z) + ⋅ ⋅ ⋅ + aiq cq (z) − m log󵄩󵄩󵄩f (z)󵄩󵄩󵄩 + O(1) 󵄩d 󵄩 = ai ⋅ c(z) − m log󵄩󵄩󵄩f (z)󵄩󵄩󵄩 + O(1). Therefore HY (m)

󵄨 󵄨 󵄩 󵄩d − ∑ log󵄨󵄨󵄨Li,z (F(z))󵄨󵄨󵄨 = SY (m, c(z)) − mHY (m) log󵄩󵄩󵄩f (z)󵄩󵄩󵄩 + O(1). i=1

Hence q

∑ log

‖f (z)‖d ‖Qj ‖ |Qj (f )(z)|

j=1

H (m)

≤

(n + 1) Y ‖F(z)‖ 󵄩 󵄩 + d(n + 1) log󵄩󵄩󵄩f (z)󵄩󵄩󵄩 ∑ log mHY (m) i=1 |Li,z (F(z))| −

‖f (z)‖d ‖Qi ‖ n+1 󵄩 󵄩 (n + 1)(2n + 1)△ log󵄩󵄩󵄩F(z)󵄩󵄩󵄩 + (max log ) + O(1). 0≤i≤q m m |Qi (f )(z)|

This proves (4.46). We now continue our proof. Integrating (4.46), applying Theorem 4.2.6 (with R0 = ∞) with ϵ = 1, and using the first main theorem and the fact that TF (r) ≤ mdTf (r) + O(1), we have q

∑ mf (r, Dj ) j=1

112 � 4 The Nevanlinna theory

≤exc −

(n + 1)(HY (m) + 1) TF (r) + d(n + 1)Tf (r) mHY (m)

‖f (reiθ )‖d ‖Qi ‖ dθ n+1 (2n + 1)(n + 1)△ TF (r) + ) + O(1) ∫ (max log 1≤i≤q m m |Qi (f )(reiθ )| 2π

d(n + 1) ≤ T (r) + d(n + 1)Tf (r) HY (m) f +

1≤j≤q

‖f (reiθ )‖d ‖Qi ‖ dθ (2n + 1)(n + 1)△ ) + O(1). ∫ (max log 1≤i≤q m |Qi (f )(reiθ )| 2π 1≤j≤q

For each j ∈ {1, . . . , q}, by Jensen’s formula, 2π

∫ log 0

2π

2π

0

0

‖f (reiθ )‖d dθ 󵄩 󵄨 󵄩 dθ 󵄨 dθ = d ∫ log󵄩󵄩󵄩f (reiθ )󵄩󵄩󵄩 − ∫ log󵄨󵄨󵄨Qj (f )(reiθ )󵄨󵄨󵄨 2π 2π |Qj (f )(reiθ )| 2π ≤ dTf (r) − Nf (r, Dj ) + O(1).

Choose m big enough such that (2n + 1)(n + 1)q△ ϵ < , m 4d

and

(n + 1)d ϵ < . HY (m) 4

Then we have q

∑ mf (r, Dj ) ≤exc d(n + 1 + ϵ)Tf (r). j=1

This proves the theorem.

4.3 The degenerate case In Theorem 4.2.3, we assumed that the map is linearly nondegenerate. In this subsection, we deal with the degenerated case, that is, the image of f : D(R0 ) → ℙn (ℂ) is contained in a proper subspace of ℙn (ℂ), where 0 < R0 ≤ ∞. We assume that the image f : D(R0 ) → ℙn (ℂ) is contained in a subspace of dimension k but not in any subspace of dimension lower than k. Without loss of generality, we assume that the subspace of dimension k that contains f (ℂ) is ℙk (ℂ). Then f : D(R0 ) → ℙk (ℂ) is linearly nondegenerate. However, the difficulty in applying the results above is that hyperplanes H1 , . . . , Hq in ℙn (ℂ) in general position may not necessarily be in general position after being restricted to ℙk (ℂ). Nochka [53, 54] (see also [11]) developed a technique to overcome this difficulty.

4.3 The degenerate case

� 113

Let n ≥ k and q ≥ n + 1. We consider the hyperplanes Hj , 1 ≤ j ≤ q, in ℙk (ℂ) given by Hj = {[x0 : ⋅ ⋅ ⋅ : xk ] ∈ ℙk (ℂ) | aj0 x0 + ⋅ ⋅ ⋅ + ajk xk = 0}, with reduced nonzero coefficient vectors aj = (aj0 , . . . , ajk ) ∈ ℂk+1 , where ℂk+1 is the dual space of ℂk+1 . ∗

∗

Definition 4.3.1. The hyperplanes H1 , . . . , Hq (or a1 , . . . , aq ) in ℙk (ℂ) are said to be in n-subgeneral position if for all 1 ≤ i0 < ⋅ ⋅ ⋅ < in ≤ q, the linear span of ai0 , . . . , ain is ℂk+1 . ∗

Given a collection of nonzero vectors A = {a1 , . . . , aq } in ℂk+1 in n-subgeneral position. We introduce the Nochka diagram and Nochka polygon of A: For a nonempty subset B of A, we associate with B a point PB = (#B, d(B)) in ℝ2 , where d(B) is the dimension of the linear span of B. The collection of the points {PB | B ⊂ A} is called the Nochka diagram of A (see Figure 4.1). In Figure 4.1 the point O = (0, 0), U = (n − k, 0), V = (n + 1, k + 1), W = , k+1 ), and X = (2n−k +1, k +1), so W is the midpoint of the segment UV , as well as ( 2n−k+1 2 2 of the segment OX. By the n-subgeneral position assumption, the points PB = (#B, d(B)) with #B ≥ n+1 lie on the line with d(B) = k +1 (i. e., they lie on the horizontal line through V and to the right of V ). On the other hand, the points PB = (#B, d(B)) with #B ≤ n + 1 lie on or above the line through U and V . ∗

Figure 4.1: Nochka diagram.

114 � 4 The Nevanlinna theory Proposition 4.3.2. Let A = {a1 , . . . , aq } be a set of vectors in ℂk+1 in n-subgeneral position. Then either σ(O, PB ) ≥ σ(O, X) for all PB = (#B, d(B)) with #B ≤ n + 1, where σ denotes the slope of the associated line segments, or there exists a uniquely determined sequence of subsets of A, ∗

A ⊃ Bs ⊃ Bs−1 ⊃ ⋅ ⋅ ⋅ ⊃ B1 ⊃ B0 = 0, with the following properties: (i) σ(PBi−1 , PBi ) < σ(PBi−1 , X), 1 ≤ i ≤ s, where σ denotes the slope of the associated line segments, and X = (2n − k + 1, k + 1). (ii) σ(O, PBi ) < σ(O, X) for 1 ≤ i ≤ s, where O = (0, 0) is the origin, and X is as above. (iii) σ(PBi−1 , PBi ) < σ(PBi , PBi+1 ), 1 ≤ i ≤ s, where we set PBs+1 = X. (iv) For 0 ≤ i ≤ s, let 𝒰i be the collection of sets B ⊂ A strictly containing Bi with #B ≤ n+1. Then σ(PBi , PBi+1 ) ≤ σ(PBi , PB ) for any B ∈ 𝒰i with strict inequality if i < s and B ∈ 𝒰i+1 . Proof. We construct the sets B1 , . . . , Bs inductively. Suppose that B0 , . . . , Bj have been constructed. Then by induction hypothesis, (i) and (ii) are satisfied for all 0 ≤ i ≤ j (these conditions are empty if j = 0), and (iii) and (iv) are satisfied for 0 ≤ i < j (these conditions are empty if j ≤ 1). If σ(PBj , PB ) ≥ σ(PBj , X) for all B ∈ 𝒰j , that is, (iv) is satisfied for i = j, then we set j = s. By (i) of the induction hypothesis we have σ(PBs−1 , PBs ) < σ(PBs−1 , X). This implies that (consider the triangle PBs−1 PBs X) σ(PBs−1 , PBs ) < σ(PBs , X), which is (iii) for the case i = j = s. We may now assume that there exists B ∈ 𝒰j such that σ(PBj , PB ) < σ(PBj , X).

(4.47)

Let σj = min{σ(PBj , PB )} B∈𝒰j

and ℳj = {B ∈ 𝒰j | σ(PBj , PB ) = σj }.

For B ∈ ℳj , we claim that d(B) < (k + 1)/2. In fact, σ(PBj , PB ) < σ(PBj , X) since B ∈ ℳj By (ii) we also have σ(O, PBj ) < σ(O, X). These conditions and the remark before the proposition imply that PB lies in the triangle OUW that appears in the Nochka diagram but not on the segment OW . Hence d(B) < (k + 1)/2. We now claim that if B and C are in ℳj , then B ∪ C is also in ℳj . First of all, since d(B∪C) ≤ d(B)+d(C) < k+1, the subgeneral position condition implies that #(B∪C) < n+1. Thus (B ∪ C) ∈ 𝒰j . Now

4.3 The degenerate case

� 115

d(B ∪ C) − d(Bj ) = d(B) + d(C) − d(B ∩ C) − d(Bj )

= d(B) − d(Bj ) + d(C) − d(Bj ) − d(B ∩ C) + d(Bj )

≤ σj {#(B) − #(Bj ) + #(C) − #(Bj ) − #(B ∩ C) + #(Bj )} = σj {#(B ∪ C) − #(Bj )},

that is, σ(PBj , PB∪C ) ≤ σj . Thus (B ∪ C) ∈ ℳj . The claim is proved. We now define Bj+1 = ⋃ B. B∈ℳj

The above claim implies that Bj+1 ∈ ℳj . We now check (i)–(iv). The proceeding argument shows that Bj+1 ∈ ℳj . By construction, σj = σ(PBj , PBj+1 ) ≤ σ(PBj , PB ) for all B ∈ 𝒰j . The inequality is strict if B is in 𝒰j+1 . Thus (iv) is verified for i = j. By assumption (4.47), σ(PBj , PBj+1 ) < σ(PBj , X), so (i) is verified for i = j+1. This, together with σ(O, PBj ) < σ(O, X) (by induction hypothesis, (ii) holds for i ≤ j), implies that the point PBj+1 is below the line OX. Hence σ(O, PBj+1 ) < σ(O, X), which is (ii) for i = j + 1. From (iv) with i = j − 1 and B = Bj+1 we have σ(PBj−1 , PBj ) < σ(PBj−1 , PBj+1 ). This implies that (by considering the triangle PBj−1 PBj PBj+1 ) σ(PBj−1 , PBj ) < σ(PBj , PBj+1 ). Thus (iii) is verified for i = j. This completes the induction step. Since the sets B0 , B1 , . . . are strictly increasing and A is a finite set, the above construction terminates after a finite number of steps, concluding the proof of the proposition. By Proposition 4.3.2, either σ(O, PB ) ≥ σ(O, X) for all PB = (#B, d(B)) with #B ≤ n + 1, or there exists a sequence B1 , . . . , Bs that appears in the proposition. If the first case occurs, then we define ω(a) = 1 for a ∈ A. Otherwise, the sequence B1 , . . . , Bs that appears in the proposition gives rise to a polygon in ℝ2 , which is called the Nochka polygon of A (see Fig. 4.1). Set Bs+1 = A. Then (B1 −B0 )∪(B2 −B1 )∪⋅ ⋅ ⋅∪(Bs −Bs−1 )∪(A−Bs ) = A is a partition of A. We now define the Nochka weights ω(a) of a ∈ A as follows: For any a ∈ A, a lies in Bi+1 − Bi for some 0 ≤ i ≤ s, and we define ω(a) = σ(PBi , PBi+1 )

(4.48)

and θ=

1

σ(PBs , X)

=

2n − k + 1 − #Bs . k + 1 − d(Bs )

(4.49)

The significance of Nochka weights is shown by the following theorem. Theorem 4.3.3 (Nochka’s theorem). Let H1 , . . . , Hq (or a1 , . . . , aq ) be hyperplanes in ℙk (ℂ) in n-subgeneral position with q ≥ 2n − k + 1. Then there exists a function ω : {1, . . . , q} → ℝ(0, 1], called a Nochka weight, and a real number θ ≥ 1, called the Nochka constant, satisfying the following properties:

116 � 4 The Nevanlinna theory (i) If j ∈ {1, . . . , q}, then 0 ≤ ω(j)θ ≤ 1. q (ii) q − 2n + k − 1 = θ(∑j=1 ω(j) − k − 1). (iii) If 0 ≠ B ⊂ {1, . . . , q} with #B ≤ n + 1, then ∑j∈B ω(j) ≤ dim L(B), where L(B) is the linear space generated by {aj | j ∈ B}. (iv) 1 ≤ (n + 1)/(k + 1) ≤ θ ≤ (2n − k + 1)/(k + 1). (v) For any real numbers E1 , . . . , Eq such that Ej ≥ 1 for 1 ≤ j ≤ q and any Y ⊂ {1, . . . , q} with 0 < #Y ≤ n + 1, there exists a subset M of Y with #M = dim L(Y ) such that {aj }j∈M is a basis for the linear space L(Y ) generated by {aj | j ∈ Y } and ω(j)

∏ Ej j∈Y

≤ ∏ Ej . j∈M

Proof. With the ω(j) and θ defined by (4.48) and (4.49), (i) immediately follows from (iii) of Proposition 4.3.2. To verify (ii), we set Bs+1 = {1, 2, . . . , q}. Then denoting σi = σ(PBi , PBi+1 ), we have q

∑ ω(j) = ∑ σi (#Bi+1 − #Bi ) j=1

0≤i≤s

=

∑ {d(Bi+1 ) − d(Bi )} + σs (q − #Bs )

0≤i≤s−1

= d(Bs ) + σs (q − #Bs ), where, by definition, σs = σ(PBs , X) =

k + 1 − d(Bs ) 1 = . 2n − k + 1 − #Bs θ

Hence d(Bs ) = k + 1 − σs (2n − k + 1 − #Bs ). Substituting yields (ii). For (iii), we consider the two cases (a) #(B ∪ Bs ) ≥ n + 1 and (b) #(B ∪ Bs ) < n + 1. If #(B ∪ Bs ) ≥ n + 1, then the subgeneral position implies k + 1 ≤ d(B ∪ Bs ).

(4.50)

By property (i) the Nochka weights satisfy σs−1 ω(a) ≤ 1. Thus ∑ ω(a) ≤ σs #B.

a∈B

By the definition of subgeneral position any set #B ≤ n + 1, say #B = n + 1 − p, satisfies k + 1 − p ≤ d(B). Thus

4.3 The degenerate case

� 117

#B ≤ d(B) + n − k. Therefore ∑ ω(a) ≤ σs (d(B) + n − k) = d(B)σs (1 +

a∈B

n−k ). d(B)

By (4.50), k + 1 ≤ d(B) + d(Bs ), so that ∑ ω(a) ≤ σs (d(B) + n − k) = d(B)σs (1 +

a∈B

n−k ) d(B)

n−k ) k + 1 − d(Bs ) n + 1 − d(Bs ) = d(B)σs k + 1 − d(Bs ) 2n − k + 1 − #Bs ≤ d(B)σs k + 1 − d(Bs ) = d(B).

≤ d(B)σs (1 +

So (iii) holds in case (a). ′ We now assume that #(B ∪ Bs ) < n + 1. Then the set Bi+1 = Bi ∪ (B ∩ Bi+1 ) contains ′ ′ Bi , and #Bi+1 ≤ #(B ∪ Bs ) < n + 1 for all i. It follows that Bi+1 is in 𝒰i , and by part (iv) of Proposition 4.3.2 σi = σ(PBi , PBi+1 ) ≤ σ(PBi , PB′ ) = i+1

d(Bi ∪ (B ∩ Bi+1 )) − d(Bi ) . #(Bi ∪ (B ∩ Bi+1 )) − #Bi

Since Bi+1 contains Bi , we have #(Bi ∪ (B ∩ Bi+1 )) − #Bi = #(B ∩ Bi+1 ) − #(B ∩ Bi ) and d(Bi ∪ (B ∩ Bi+1 )) − d(Bi ) = d(B ∩ Bi+1 ) − d(B ∩ Bi ). Thus σi ≤

d(B ∩ Bi+1 ) − d(B ∩ Bi ) . #(B ∩ Bi+1 ) − #(B ∩ Bi )

We can now easily estimate the sum of Nochka weights: ∑ ω(a) ≤ ∑ σi {#(B ∩ Bi+1 ) − #(B ∩ Bi )}

a∈B

0≤i≤s

≤ ∑ (d(B ∩ Bi+1 ) − d(B ∩ Bi )) 0≤i≤s

= d(B ∩ Bs+1 ) = d(B). This completes the proof of (iii). To verify (iv), since PBs lies below the line OX in the Nochka diagram, σs = σ(PBs , X) > σ(O, X) = (k + 1)/(2n − k + 1). On the other hand, PBs lies below the triangle OUW , and thus σ(PBs , X) < σ(U, X) = (k + 1)/(n + 1). This proves (iv).

118 � 4 The Nevanlinna theory Finally, we prove (v). Without loss of generality, we assume that 1 ≤ Eq ≤ Eq−1 ≤ ⋅ ⋅ ⋅ ≤ E1 . Define an increasing sequence of subsets of Y as follows: Let i1 = min{i | i ∈ Y } and I1 = {i ∈ Y | ai is a multiple of ai1 }. If Y − I1 ≠ 0, then choose i2 ∈ Y − I1 such that i2 = min{i | i ∈ Y − I1 } and I2 = {i ∈ Y | ai ∈ linear span of ai1 , ai2 }. Inductively, if Ij−1 is defined and if Y − Ij−1 ≠ 0, then ij = min{i | i ∈ Y − Ij−1 } and Ij = {i ∈ Y | ai ∈ linear span of ai1 , . . . , aij }. This process stops at Ip with p = the dimension of L(Y ). It is clear that Ip ⊃ Ip−1 ⊃ ⋅ ⋅ ⋅ and ip ≥ ip−1 ≥ ⋅ ⋅ ⋅ ≥ i1 . Let M = {i1 , . . . , ip }. Then, by construction, the set {aj }j∈M is a basis for L(Y ). Set I0 = 0. Then Y = ⋃1≤j≤p (Ij − Ij−1 ) is a disjoint union. Since Eq ≤ ⋅ ⋅ ⋅ ≤ E1 , by construction we have Eij = max Ei . i∈Ij −Ij−1

Thus ω(j)

∏ Ej j∈Y

α

∏ Eiω(i) ≤ ∏ Ei j ,

≤ ∏

1≤j≤p i∈Ij −Ij−1

j

1≤j≤p

where αj = ∑i∈Ij −Ij−1 ω(i). Since the sets Ij are increasing, for any 1 ≤ r ≤ p, (iii) implies that ∑ αj = ∑

1≤j≤r

∑ ω(i) = ∑ ω(i) ≤ d(Ir ) = r.

1≤j≤r i∈Ij −Ij−1

i∈Ir

(4.51)

It remains to show that α

∏ Ei j ≤ ∏ Eij .

1≤j≤p

j

1≤j≤p

This is easily verified by the induction on p. For p = 1, by (iii), α1 ≤ 1, and since E1 ≥ 1, α we trivially have E1 1 ≤ E1 . Assume that the inequality holds for p = k. Since αk+1 ≤ k + 1 − ∑1≤j≤k αj by (4.51), we have α

∏ Ei j ≤ ( ∏ (

1≤j≤k+1

j

1≤j≤k

Eij

Eik+1

αj

) )Eik+1 . k+1

Since ip ≥ ⋅ ⋅ ⋅ ≥ i1 , we have 1 ≤ Eij /Eik+1 for 1 ≤ j ≤ k, and the induction hypothesis implies that (∏( 1≤j≤k

Eij

Eik+1

αj

) )Eik+1 ≤(∏ k+1

1≤j≤k

Eij

Eik+1

)Eik+1 = ∏ Eij . k+1 1≤j≤k+1

4.3 The degenerate case

� 119

The combination of the last two inequalities completes the proof of (v). Theorem 4.3.4 (Degenerated SMT). Let f = [f0 : ⋅ ⋅ ⋅ : fn ] : D(R0 ) → ℙn (ℂ) be a holomorphic map whose image is contained in some k-dimensional subspace but not in any subspace of dimension lower than k, where 0 < R0 ≤ ∞. Let Hj , 1 ≤ j ≤ q, be hyperplanes in ℙn (ℂ) located in general position. Assume that f (D(R0 )) ⊄ Hj for 1 ≤ j ≤ q. Then for every ϵ > 0, we have q

∑ mf ,Hj (r) + ( j=1

n+1 2n − k + 1 )N(Rf , r) ≤ (2n − k + 1)Tf (r) + ( )S(r) k+1 k+1 +(

k(2n − k + 1) )ϵ log r 2

for 0 < r < R0 , where N(Rf , r) is the ramification term defined in the proof, and S(r) is the same as in Theorem 4.2.3. Proof. Without loss of generality, we may assume that f (D(R0 )) ⊂ ℙk (ℂ). So f : D(R0 ) → ℙk (ℂ) is a linearly nondegenerate holomorphic map. We also assume that q ≥ 2n − k + 1. Denote Ĥ j = Hj ∩ ℙk (ℂ). Then Ĥ j are hyperplanes in ℙk (ℂ) located in n-subgeneral position. Recall that the Weil function of f is defined as λH (f (z)) = log

‖f(z)‖‖a‖ ≥ 0. |⟨f(z), a⟩|

Since H1 , . . . , Hq are in general position in ℙn (ℂ), for each z ∈ ℂ, there exist indices i(z, 0), . . . , i(z, n) ∈ {1, . . . , q} such that q

n

j=1

l=0

∑ ω(j)λHj (f (z)) ≤ ∑ ω(i(z, l))λHi(z,l) (f (z)). λ

(4.52)

By Theorem 4.3.3 with El = e Hi(z,l) , 0 ≤ l ≤ n, there is a subset M of Y = {i(z, 0), . . . , i(z, n)} with #M = k + 1 such that {Ĥ i(z,j) | i(z, j) ∈ M} is linearly independent and n

∏e

(f (z))

̂

ω(i(z,l))λĤ

i(z,l)

(f (z))

≤ ∏ e

λĤ

i(z,j)

(f (z))

i(z,j)∈M

l=0

.

Thus n

∑ ω(i(z, l))λĤ

l=0

i(z,l)

(f (z)) ≤

k

∑ λĤ

i(z,j)∈M

i(z,j)

(f (z)) ≤ max ∑ λĤ (f (z)), γ∈Γ

l=0

γ(l)

where Γ is the set of all maps γ : {0, . . . , k} → {1, . . . , q} such that Ĥ γ(0) , . . . , Ĥ γ(k) are linearly independent. Combining this with (4.52) gives us

120 � 4 The Nevanlinna theory 2π

q

k

∑ ω(j)mf (Hj , r) ≤ ∫ max ∑ λĤ (f (reiθ )) j=1

0

γ∈Γ

γ(l)

l=0

dθ + O(1). 2π

Applying Theorem 4.2.6 yields 2π

k

∫ max ∑ λĤ (f (reiθ )) γ∈Γ

0

l=0

γ(l)

dθ k(k + 1) + N(Rf , r) ≤ (k + 1)Tf (r) + S(r) + ϵ log r, 2π 2

where N(Rf , r) := NW (r, 0). Hence q

∑ ω(j)mf (Hj , r) + N(Rf , r) ≤ (k + 1)Tf (r) + S(r) + j=1

k(k + 1) ϵ log r. 2

By Theorem 4.3.3, recalling that mf (r, Hj ) ≤ Tf (r) + O(1), this inequality gives q

q

q

j=1

j=1

j=1

∑ mf (r, Hj ) = ∑(1 − θω(j))mf (r, Hj ) + ∑ θω(j)mf (r, Hj ) q

≤ ∑(1 − θω(j))mf (r, Hj ) + θ(k + 1)Tf (r) − θN(Rf , r) + θS(r) j=1

+θ q

k(k + 1) ϵ log r 2

≤ ∑(1 − θω(j))Tf (r) + θ(k + 1)Tf (r) − ( j=1

+θ

k(k + 1) ϵ log r 2

= {q − θ( ∑ ω(j) − k − 1)}Tf (r) − ( 1≤j≤q

k(k + 1) +θ ϵ log r 2 = (2n − k + 1)Tf (r) − ( +(

n+1 )N(Rf , r) + θS(r) k+1

n+1 )N(Rf , r) + θS(r) k+1

n+1 2n − k + 1 )N(Rf , r) + ( )S(r) k+1 k+1

k(2n − k + 1) )ϵ log r. 2

Corollary 4.3.5. Let Hj , 1 ≤ j ≤ q, be hyperplanes in ℙn (ℂ) in general position, and let q q ≥ 2n + 1. Then every holomorphic map f : ℂ → ℙn (ℂ) − ⋃j=1 Hj is constant. q

Proof. Assume that f : ℂ → ℙn (ℂ) − ⋃j=1 Hj is not constant. Then the image of f is contained in some k-dimensional subspace but not in any subspace of dimension lower than k with 1 ≤ k ≤ n. Applying Theorem 4.3.4 with R0 = ∞, we get

4.4 Holomorphic curves through Ahlfors’ negative curved method

� 121

q

∑ mf ,Hj (r) ≤exc (2n − k + 1)Tf (r) + O(log+ Tf (r)) + ϵ log r. j=1

Since the image of f omits Hj , we have Nf (r, Hj ) = 0, so by the first main theorem, mf (r, Hj ) = Tf (r) + O(1) for 1 ≤ j ≤ 2n + 1. Hence we get (2n + 1)Tf (r) ≤exc (2n − k + 1)Tf (r) + O(log+ Tf (r)) + ϵ log r, that is, kTf (r) ≤exc O(log+ Tf (r)) + ϵ log r, which gives a contradiction.

4.4 Holomorphic curves through Ahlfors’ negative curved method We follow the method by Ahlfors to extend the theory of algebraic curves (see Chapter 2) to holomorphic curves. The associated maps Let D(R) = {z | |z| < R} ⊂ ℂ with 0 < R ≤ +∞, and let f : D(R) → ℙn (ℂ) be a linearly nondegenerate map, that is, the image of f is not contained in any proper subspace of ℙn (ℂ). Let f : D(R) → ℂn+1 − {0} be a reduced representation of f . Consider the holomorphic map k+1

Fk = f ∧ f′ ∧ ⋅ ⋅ ⋅ ∧ f(k) : DR → ⋀ ℂn+1 . Evidently, Fn+1 ≡ 0. Throughout this section, we assume that f is linearly nondegenerate, so Fk ≢ 0 for 0 ≤ k ≤ n. The map Fk = ℙ(Fk ) : D(R) → ℙ(⋀k+1 ℂn+1 ) = ℙNk , where (n+1)! Nk = (k+1)!(n−k)! − 1 and ℙ is the natural projection, is called the kth associated map. We can express them by using the Wronskians. For holomorphic functions f0 , f1 , . . . , fn , we define the Wronskian W (f0 , f1 , . . . , fn ) by W (f0 , f1 , . . . , fn ) := det(fj(s) , 0 ≤ j, s ≤ n). Let {e0 , e1 , . . . , en } be the standard basis of ℂn+1 . Let f = (f0 , . . . , fn ) be a reduced representation of f . Then Fk = ‖Fk ‖2 =

∑

W (fj0 , fj1 , . . . , fjk )ej0 ∧ ⋅ ⋅ ⋅ ∧ ejk ,

(4.53)

∑

󵄨󵄨 󵄨2 󵄨󵄨W (fj0 , fj1 , . . . , fjk )󵄨󵄨󵄨 .

(4.54)

0≤j0 0 such that #Ix (c) ≤ m − p for all x ∈ ℙ(⋀p ℂn+1 ). Proof. Take x ∈ ℙ(⋀p ℂn+1 ). Let Ix be the set defined in Lemma 4.4.3. There exists cx > 0 such that ‖x; Hj ‖2 > cx for all j ∈ ℕ[1, q] − Ix . There exists an open neighborhood Ux of x in ℙ(⋀p ℂn+1 ) such that ‖y; Hj ‖2 > cx for all y ∈ Ux and j ∈ ℕ[1, q] − Ix . Since ℙ(⋀p ℂn+1 ) is compact, there are only finitely many Ux1 , . . . , Uxs whose union covers ℙ(⋀p ℂn+1 ). Define c = min{cx1 , . . . , cxs } > 0. Take y ∈ ℙ(⋀p ℂn+1 ). Then there exists λ ∈ ℕ[1, s] such that y ∈ Uxλ . Take j ∈ Iy (c). Assume that j ∈ ̸ Ixλ . Then ‖y; Hj ‖2 > cxλ ≥ c ≥ ‖y; Hj ‖2 , which is impossible. Hence Iy (c) ⊂ Ixλ and #Iy (c) ≤ #Ixλ ≤ m − p. Theorem 4.4.5 (Product-to-sum estimate). Let H1 , . . . , Hq (or a1 , . . . , aq ) be hyperplanes in ℙn (ℂ) in m-general position. Let ω(j), j = 1 . . . , q, be the Nochka weights. For any 1/q ≤ λk ≤ 1/(n − k), there exists a constant Ck > 0, which depends only on k and the given hyperplanes, such that q

∏[( j=1 q

ϕk+1 (Hj ) ϕk (Hj )

ω(j)

)

on DR0 − ⋃j=1 {ϕk (Hj ) = 0}.

1 ] log(μ/ϕk (Hj ))

λk

q

≤ Ck ∑ j=1

ϕk+1 (Hj )

ϕk (Hj ) log(μ/ϕk (Hj ))

4.4 Holomorphic curves through Ahlfors’ negative curved method

� 129

Proof. By Lemma 4.4.4 there exists a constant c > 0 such that #Ix (c) ≤ m − k for all x ∈ ℙ(⋀k ℂn+1 ). Fix z ∈ ℂ and let x = Fk (z). There exists a subset A ⊂ {1, . . . , q} such that Ix (c) ⊂ A with #A = m + 1. By Nochka’s theorem there exists B ⊂ A with #B = n + 1 such that aj , j ∈ B, are linearly independent and ∏(

ω(j)

ϕk+1 (Hj )

)

ϕk (Hj )

j∈A

ϕk+1 (Hj )

≤∏

ϕk (Hj )

j∈B

.

For j ∈ ̸ A (hence j ∈ ̸ Ix (c)), ϕk+1 (Hj )/ϕk (Hj ) ≤ 1/ϕk (Hj ) ≤ 1/c, we get q

∏( j=1

ϕk+1 (Hj ) ϕk (Hj )

ω(j)

)

≤

c

1 ∑j∈A̸

∏(

ϕk+1 (Hj ) ϕk (Hj )

j∈A

ω(j)

)

≤

c

1 ∑j∈A̸

∏ j∈B

ϕk+1 (Hj ) ϕk (Hj )

.

Now denote Ix (d, B) = {j ∈ B | ‖x; Hj ‖2 ≤ d}. Since Hj , j ∈ B, is in general position, by Lemma 4.4.4 (with m = n) we indeed get #(Ix (d, B)) ≤ n − k for some d > 0. Hence there exists a subset I ⊂ B such that #I = n − k. Thus #(B − I) = n + 1 − (n − k) = k + 1, and for j ∈ B − I, ϕk+1 (Hj )/ϕk (Hj ) ≤ 1/ϕk (Hj ) ≤ 1/d, so we get q

∏( j=1

ϕk+1 (Hj ) ϕk (Hj )

ω(j)

)

1

≤

c

∑j∈A̸

=

c

∑j∈A̸

1

≤C∏ j∈I

∏

ϕk+1 (Hj )

j∈B

ϕk (Hj )

ϕk+1 (Hj )

∏

ϕk (Hj )

j∈B−I

ϕk+1 (Hj ) ϕk (Hj )

∏ j∈I

ϕk+1 (Hj ) ϕk (Hj )

.

Hence, noting that 1/(log(μ/ϕk (Hj )) ≤ (1/ log μ) := c,̃ we get q

∏[(

ϕk+1 (Hj )

j=1

ϕk (Hj )

q

≤ ∏(

ω(j)

ϕk+1 (Hj )

j=1

ϕk (Hj )

≤ C c̃q−n+k ∏ j∈I

= Ĉ ∏ j∈I

1 ] log(μ/ϕk (Hj ))

)

ω(j) q

)

∏ j=1

ϕk+1 (Hj ) ϕk (Hj )

ϕk+1 (Hj )

1 log(μ/ϕk (Hj ))

∏ j∈I

1 log(μ/ϕk (Hj ))

ϕk (Hj ) log(μ/ϕk (Hj ))

.

130 � 4 The Nevanlinna theory On the other hand, we have ∏( j∈I

1/(n−k)

ϕk+1 (Hj )

ϕk (Hj ) log(μ/ϕk (Hj )

)

≤

ϕk+1 (Hj ) 1 ∑ n − k j∈I ϕk (Hj ) log(μ/ϕk (Hj )

≤

q ϕk+1 (Hj ) 1 , ∑ n − k j=1 ϕk (Hj ) log(μ/ϕk (Hj )

since the arithmetic mean majorizes the geometric mean. Thus q

∏[( j=1

ϕk+1 (Hj ) ϕk (Hj )

≤ Ĉ λk ∏[( j∈I

≤ Ĉ λk ∏( j∈I

≤

ω(j)

)

λk

1 ] log(μ/ϕk (Hj )

ϕk+1 (Hj ) ϕk (Hj )

λk

)

1 ] log(μ/ϕk (Hj )

ϕk+1 (Hj )

1/(n−k)

ϕk (Hj ) log(μ/ϕk (Hj )

)

q ϕk+1 (Hj ) Ĉ λk , ∑ n − k j=1 ϕk (Hj ) log(μ/ϕk (Hj )

and the lemma is proved. Let f : D(R0 ) → ℙn (ℂ) be a linearly nondegenerate holomorphic map with R0 ≤ ∞. Let H1 , . . . , Hq hyperplanes in ℙn (ℂ) located in m-subgeneral position. We use f to construct a pseudo-metric on D(R0 ). We write Ωk =

√−1 a (z)dz ∧ d z,̄ 2π k

(4.68)

where Ωk is defined in (4.55). Let q

σk = Ck−1 ∏[( j=1

ϕk+1 (Hj )

ϕk (Hj )(N − log ϕk (Hj ))

ω(j) λk

) 2

] ⋅ ak ,

where Ck is the positive constant in the product-to-sum estimate above, λk = 1/[n − k + 2q(n − k)2 /N], and N ≥ 1. We take the geometric mean of σk and define Γ=

√−1 n−1 βn /λk c∏σ dz ∧ d z,̄ 2π k=0 k λ−1

n−1 −1 k βn where βn = 1/ ∑n−1 k=0 λk and c = 2(∏k=0 λk ) . Let

Γ=

√−1 h(z)dz ∧ d z.̄ 2π

4.4 Holomorphic curves through Ahlfors’ negative curved method

� 131

Then n−1

h(z) = (c ∏ Ck k=0

k

βn q

q

β

− λn

) ∏( j=1

βn qλ

n−1

ak k

1 ) ∏[∏ ]. 2βn ϕ0 (Hj )ω(j) j=1 k=0 (N − log ϕk (Hj ))

(4.69)

Theorem 4.4.6. For q ≥ 2m − n + 2 and q

2q/N
0 j=1

q

and θ > 0, so (∑j=1 ω(j) − (n + 1)) > 0. This and the choice of N give us dd c log h(z) ≥

√−1 h(z)dz ∧ d z.̄ 2π

We use this alternative “negative curvature method” to obtain the following theorem again (see Corollary 4.3.5). Theorem 4.4.7. Let Hj , 1 ≤ j ≤ q, be hyperplanes in ℙn (ℂ) in general position, and let q q ≥ 2n + 1. Then every holomorphic map f : ℂ → ℙn (ℂ) − ⋃j=1 Hj is constant.

4.4 Holomorphic curves through Ahlfors’ negative curved method

� 133

Proof. Let h be defined above. By the Ahlfors–Schwarz lemma we have 2

2R ). h(z) ≤ ( 2 R − |z|2 Letting R → ∞, we have h(z) ≡ 0 on ℂ. Hence f is constant. A complex manifold M is said to be Brody hyperbolic if every holomorphic map f : q ℂ → M is constant. From the above we know that M := ℙn (ℂ)−⋃j=1 Hj , where H1 , . . . , Hq are in general position and q ≥ 2n + 1, is Brody hyperbolic. Indeed, it is also complete Kobayashi hyperbolic and is hyperbolically imbedded in ℙn (ℂ). For the notations and details, see Lang [47] or Ru [76]. Theorem 4.4.8 (See Lang [47] or Ru [76]). Let H1 , . . . , Hq be hyperplanes in ℙn (ℂ), located q in general position. If q ≥ 2n + 1, then X = ℙn (ℂ) − ⋃j=1 Hj is complete hyperbolic and hyperbolically imbedded in ℙn (ℂ). Hence if D ⊂ ℂ is the unit disc and Φ is a subset of Hol(D, X), then Φ is relatively locally compact in Hol(D, ℙn (ℂ)), that is, given a sequence {fn } in Φ, there exists a subsequence that converges uniformly on every compact subset of D to an element of Hol(D, ℙn (ℂ)). Denote |Fk (H)| := ‖Fk ⌊a‖ whose formula is given in (4.58). As a consequence of Theorem 4.4.6, we get the following: Lemma 4.4.9. Let F : D(R) → ℙn (ℂ) be a linearly nondegenerate holomorphic map. q Let {Hj }j=1 be a set of hyperplanes in ℙn (ℂ) in m-subgeneral position, and let ω(j) be their Nochka weights. Let F be a reduced representation of F. If q > 2m − n + 1 and , then there exists a constant C > 0 such that N > ∑q 2qn(n+2) ω(j)−(n+1) j=1

2q

χ

(‖F‖

4

q

N ‖Fn ‖1+ N ∏j=1 ∏n−1 p=0 |Fp (Hj )|

q

∏j=1 |F(Hj )|ω(j)

q

where χ = ∑j=1 ω(j) − (n + 1) − β /λp

Proof. We will calculate apn

1/λ ap p

2q 2 (n N

1

2 2R )(z) ≤ C( 2 ) 2 R − |z|

2q

n(n+1)+ N ∑ns=0 s2

+ 2n − 1).

that appeared in (4.69). By (4.56)

=(

‖Fp−1 ‖2 ‖Fp+1 ‖2 ‖Fp ‖4

(n−p)+2q(n−p)2 /N

)

.

Therefore n−1

1/λp

∏ ap p=0

2

= ‖F0 ‖−2(n+1)−(n +2n−1)4q/N ‖F1 ‖8q/N ⋅ ⋅ ⋅ ‖Fn−1 ‖8q/N ‖Fn ‖2+4q/N .

Since F0 = F and ϕp (H) =

|Fp (H)|2 , ‖Fp ‖2

by (4.69) and the inequality

,

134 � 4 The Nevanlinna theory

h(z) ≤ (

2

2R ) 2 R − |z|2

we get 󵄩χ 󵄩󵄩 󵄩󵄩F(z)󵄩󵄩󵄩

(‖F1 (z)‖ ⋅ ⋅ ⋅ ‖Fn−1 (z)‖)4q/N ‖Fn (z)‖1+2q/N

q

∏j=1 (|F(Hj )(z)|ω(j) ∏n−1 p=0 (N − log ϕp (Hj )(z)))

≤ C(

1/βn

2R ) R2 − |z|2

Set K := sup0 0. 𝜕(x, y) A Therefore by the inverse function theorem the inverse L−1 exists (locally). Let x̃ := x∘L−1 . We prove that this new parameterization is isothermal. Indeed, by the chain rule, (

x̃ u (x ) )=( 1 u x̃ v (x1 )v

(x2 )u x ) ( x1 ) . (x2 )v xx2

Hence t

(

x̃ u x̃ 1 + ‖p‖2 ) ( u ) = J −1 ( x̃ v x̃ v p⋅q

p⋅q 1 A −1 t ( 2 ) (J ) = 1 + ‖q‖ 0 det J

Hence we get x̃ u ⋅ x̃ u = x̃ v ⋅ x̃ v =

A , det J

x̃ u ⋅ x̃ v = 0.

0 ). 1

138 � 5 Gauss maps of minimal surfaces in ℝm Therefore the new parameterization x̃ is isothermal. Let M be a minimal surface immersed in ℝm . Similarly in Section 3.1, by Proposition 3.2.2, M becomes a Riemann surface with local isothermal coordinates (u, v) by letting z = u + iv. Let x : M → ℝm be an immersion. Let {U, ϕ = (u, v)} be a local isothermal coordinate. We write x(u, v) = (x ∘ ϕ−1 )(u, v), which is a local parameterization for M. The induced metric on M ⊂ ℝm from the standard Euclidean metric on ℝm is ds2 = λ2 (du2 + dv2 ), where λ = xu ⋅ xu = xv ⋅ xv . We claim that △ds2 x = 2HN,

(5.2)

where △ds2 is the Laplacian operator with respect to this induced metric given by △ds2 :=

1 𝜕 𝜕 4 𝜕2 . ( + ) = λ2 𝜕u2 𝜕v2 λ2 𝜕z𝜕z̄

To prove this equation, notice that x(u, v) is isothermal, so xu ⋅ xu = xv ⋅ xv and xu ⋅ xv = 0. By differentiating these equations with respect to u and v we get xuu ⋅ xu = xuv ⋅ xv

and xvu ⋅ xv = −xvv ⋅ xu .

Hence (xuu + xvv ) ⋅ xu = xuv ⋅ xv − xvu ⋅ xv = 0. Similarly, we can get (xuu +xvv )⋅xv = 0. From this we conclude that △ds2 x is proportional to N. On the other hand, for every normal vector, H(N) =

h11,N + h22,N 2λ2

1 = (△ds2 x) ⋅ N. 2

This proves (5.2). Therefore we get the following result. Proposition 5.1.3. A surface M in ℝm is minimal if only if its isothermal parameterization x is harmonic. Write x(u, v) = (x1 (u, v), . . . , xm (u, v)). Let ϕk :=

𝜕xk 𝜕z

= 21 ( 𝜕uk − i 𝜕x

𝜕xk ) 𝜕v

for k = 1, . . . , m.

Then the condition that M is minimal implies that x is harmonic, and hence

Thus

𝜕ϕk 𝜕z̄

= 0, so ϕk is holomorphic for k = 1, 2, . . . , m. Also, m

m

k=1

k=1

4 ∑ ϕ2k = ∑ (

2

2

m m 𝜕xk 𝜕x 𝜕x 𝜕x ) − ∑ ( k ) − 2i ∑ k k 𝜕u 𝜕v 𝜕u 𝜕v k=1 k=1

𝜕2 x 𝜕z𝜕z̄

= 0.

5.1 Minimal surfaces in ℝm

�

139

= ‖xu ‖2 − ‖xv ‖2 − 2ixu ⋅ xv = 0 and m

m

k=1

k=1

4 ∑ |ϕk |2 = ∑ (

2

2

m 𝜕x 𝜕xk ) + ∑ ( k ) = ‖xu ‖2 + ‖xv ‖2 ≠ 0 𝜕u 𝜕v k=1

if M is regular. The differentials αk := ϕk dz, 1 ≤ k ≤ 3, are independent of the local coordinate z and hence are globally defined differential forms on M. Thus we obtain the following: Theorem 5.1.4. Let M be an open Riemann surface, and let ω1 , ω2 , . . . , ωm be holomorphic forms on M having no common zeros and no real periods and locally satisfying the identity f12 + f22 + ⋅ ⋅ ⋅ + fm2 = 0 for holomorphic functions fi with ωi = fi dz. Set z

xi = 2 Re ∫ ωi z0

for an arbitrary fixed point z0 of M. Then the surface x = (x1 , . . . , xm ) : M → ℝm is a minimal surface immersed in ℝm . Definition 5.1.5. The generalized Gauss map of M ⊂ ℝm is defined as G : M → ℙm−1 (ℂ),

G = ℙ(

𝜕x ) = [ϕ1 : ⋅ ⋅ ⋅ : ϕm ]. 𝜕z

Note that the definition is independent of the choice of z. If M is minimal, then G is a holomorphic map. Moreover, since ϕ21 + ⋅ ⋅ ⋅ + ϕ2m = 0, the image of G is contained in the quadric 2 Qm−1 = {[w1 : ⋅ ⋅ ⋅ : wm ] | w12 + ⋅ ⋅ ⋅ + wm = 0} ⊂ ℙm−1 (ℂ).

The induced metric on M under the isothermal coordinate (u, v) is 󵄨󵄨 𝜕x 󵄨󵄨2 󵄨󵄨 𝜕x 󵄨󵄨2 󵄨 󵄨 󵄨 󵄨 ds2 = (xu ⋅ xu )(du2 + dv2 ) = 2(󵄨󵄨󵄨 1 󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨 m 󵄨󵄨󵄨 )|dz|2 , 󵄨󵄨 𝜕z 󵄨󵄨 󵄨󵄨 𝜕z 󵄨󵄨 that is, ̃ 2 |dz|2 , ds2 = 2‖G‖ where G̃ is any reduced representation of G. Its Gauss curvature by (3.6) is

140 � 5 Gauss maps of minimal surfaces in ℝm

K =−

2 𝜕2 ̃ = 1 (∑󵄨󵄨󵄨gi g ′ − gj g ′ 󵄨󵄨󵄨2 ). log ‖G‖ i󵄨 󵄨 j ̃ 2 𝜕z𝜕z̄ ̃ 6 ‖G‖ ‖G‖ i mk(n− k−1 )+2n−k+1. 2 By taking the universal cover of M if necessary we can assume that M is simply connected. Then the uniformization theorem implies that M is conformally equivalent either to ℂ or to the unit disc Δ. By Theorem 4.3.4 we know that a k-nondegenerate holomorphic map from ℂ to ℙn (ℂ) cannot omit more than 2n − k + 1 hyperplanes in general position. Hence, noticing that mk(n− k−1 )+2n−k +1 ≥ 2n−k +1, G must be k-degenerate if 2 M is conformally equivalent to ℂ, which contradicts with our assumption. So it remains

142 � 5 Gauss maps of minimal surfaces in ℝm to prove the theorem in the case that M is conformally equivalent to the unit disc Δ. We assume that M is the unit disc Δ. Since G is k-nondegenerate, the image of G is contained in some k-dimensional projective subspace of ℙn (ℂ) but not in any subspace of dimension lower than k. Therefore by regarding ℙk (ℂ) ⊂ ℙn (ℂ) the map G : Δ → ℙk (ℂ) is linearly nondegenerate. Let q H̃ j := Hj ∩ ℙk (ℂ), 1 ≤ j ≤ q. Then these hyperplanes {H̃ j }j=1 are in n-subgeneral position q

in ℙk (ℂ) since {Hj }j=1 are, by assumption, in general position in ℙn (ℂ). We still denote H̃ j as Hj . Hence it is reduced to the case that G : Δ → ℙk (ℂ) is linearly nondegenerate and the hyperplanes Hj , 1 ≤ j ≤ q, in ℙk (ℂ) are located in n-subgeneral position. Let G̃ = (g0 , g1 , . . . , gk ) be a reduced representation of G, and let aj := (aj0 , aj1 , . . . , ajk ) be the unit vectors associated with Hj , 1 ≤ j ≤ q. From the discussion in the last chapter, G̃ k ≢ 0, and none of the ‖G̃ s ⌊aj ‖, 0 ≤ s ≤ k, 1 ≤ j ≤ q, vanishes identically, where, according to (4.58), 󵄩󵄩 ̃ 󵄩2 󵄩󵄩Gs ⌊aj 󵄩󵄩󵄩 =

󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨 󵄨󵄨 ∑ ajj0 W (gj0 , gj1 , . . . , gjs )󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨 0≤j1 1. mk(2n − k + 1)

q

Therefore ∑j=1 ω(j) − k − 1 −

mk (k 2

+ 1) > 0. Set

5.2 Value distributions of the (generalized) Gauss maps of complete minimal surfaces in ℝm q

χ : = ∑ ω(j) − (k + 1) − j=1

�

143

2q 2 (k + 2k − 1), N

2q k 2 m 1 ( k(k + 1) + ∑ s ). χ 2 N s=0

λ:=

Choose N such that the following inequality holds: q

2m q

mk (k 2

q

mk 2q ∑j=1 ω(j) − k − 1 − 2 (k + 1) < , < N m ∑ks=0 s2 + k 2 + 2k − 1 + m ∑ks=0 s2 + k 2 + 2k − 1

∑j=1 ω(j) − k − 1 −

+ 1)

which implies that 0 < λ < 1,

4m > 1. Nχ(1 − λ)

Write ω = hdz, where h is a nowhere vanishing holomorphic function and define a new flat metric q

∏j=1 |aj0 g0 + ⋅ ⋅ ⋅ + ajk gk |ω(j)

2

dσ = (

|W (g0 , . . . , gk )|

2q

1+ N

4 q N ∏j=1 (∏k−1 s=0 |ψjs |)

2m (1−λ)χ

2

|h| 1−λ |dz|2

)

on the set q k−1

M ′ := Δ\{p ∈ Δ | either W (g0 , . . . , gk )(p) = 0 or (∏ ∏ |ψjs |)(p) = 0}. j=1 s=0

Here W (g0 , . . . , gk ) is the Wronskian determinant of G̃ = (g0 , . . . , gk ). Fix a point p0 ∈ M ′ . By Lemma 3.3.1 there exists a local diffeomorphism Ψ of a disk ΔR = {w ∈ ℂ : |w| < R} (0 < R ≤ ∞) onto an open neighborhood of p0 with Ψ(0) = p0 such that Ψ is a local isometry. In addition, there exists a point a0 with |a0 | = 1 such that the Ψ-image Γa0 of the curve La0 = {w = a0 t : 0 < t < R} is divergent in M ′ . Since G is not constant, R < ∞. We claim that the Ψ-image Γa0 is indeed divergent to the boundary of Δ. To this end, we assume the contrary that the curve Γa0 is divergent to a point z0 that satisfies either G̃ k (z0 ) = 0 or ψjs (z0 ) = 0 for some s and j. Set q

Λ=(

∏j=1 |aj0 g0 + ⋅ ⋅ ⋅ + ajk gk |ω(j) 2q

q

m (1−λ)χ 4

N |W (g0 , . . . , gk )|1+ N ∏j=1 (∏k−1 s=0 |ψjs |)

)

.

Then, also noticing that z0 is a pole of Λ with the multiplicities at least δ0 = and using the fact that Ψ is an isometry,

4m (> Nχ(1−λ0 )

1)

144 � 5 Gauss maps of minimal surfaces in ℝm 1 󵄨 1 󵄨 R = length of La0 = ∫ Λ(z)󵄨󵄨󵄨h(z)󵄨󵄨󵄨 1−λ |dz| ≥ c ∫ |dz| = ∞. |z − z0 |δ0 Γa0

Γa0

This gives a contradiction. Therefore Γa0 = Ψ(La0 ) is divergent to the boundary of Δ. ̃ 2m |ω|2 , Our goal now is to show that Γa0 has finite length in the (original) metric ‖G‖ 2m 2 ̃ |ω| . To do so, we use the contradicting the completeness of the (original) metric ‖G‖ isometry property of Ψ, that is, if we let z = Ψ(w), then q

|dw| = dσ = (

∏j=1 |aj0 g0 + ⋅ ⋅ ⋅ + ajk gk |ω(j) 2q 1+ N

|W (g0 , . . . , gk )|

q

∏j=1 (∏k−1 s=0 |ψjs |)

m (1−λ)χ 4 N

)

1

|h| 1−λ |dz|.

(5.5)

To change the coordinate z into w in the right-hand side of (5.5), we introduce the following functions defined on {w : |w| < R}: fs (w) := gs (Ψ(w)), 0 ≤ s ≤ k, and f := [f0 : f1 : ⋅ ⋅ ⋅ : fk ], that is, f (w) = G(Ψ(w)). For 1 ≤ j ≤ q and 0 ≤ s ≤ k, we let φjs :=

∑ ajt W (ft , fi1 , . . . , fis ),

t =i̸ 1 ,...,is

where (i1 , . . . , is ) is the index in the definition of ψjs in (5.4). Denote by Fs (w) and G̃ s (z) the sth associated maps of f and G, respectively. Then it is obvious from the definition that Fs (w) = G̃ s (z)(

s(s+1)/2

dz ) dw

0 ≤ s ≤ k.

,

In particular, noticing that |W (f0 , . . . , fk )| = ‖Fk ‖, we have 󵄨k(k+1)/2 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 . 󵄨󵄨W (f0 , . . . , fk )󵄨󵄨󵄨 = 󵄨󵄨󵄨W (g0 , . . . , gk )󵄨󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨󵄨 dw 󵄨󵄨󵄨 Thus, if we let F := (f0 , . . . , fk ), then (5.5) becomes q

|dw| = (

∏j=1 |aj0 f0 + ⋅ ⋅ ⋅ + ajk fk |ω(j) 2q

q

m (1−λ)χ 4

N |W (f0 , . . . , fk )|1+ N ∏j=1 (∏k−1 s=0 |φjs |) m((1+

2q k(k+1) 2q k−1 ) + s(s+1)) ∑

)

N N s=0 2 󵄨󵄨 dz 󵄨󵄨 (1−λ)χ 1 󵄨 󵄨󵄨 × 󵄨󵄨󵄨 |h| 1−λ |dz| 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 m λ q 󵄨 ∏j=1 |aj0 f0 + ⋅ ⋅ ⋅ + ajk fk |ω(j) (1−λ)χ 󵄨󵄨 1 󵄨󵄨 dz 󵄨󵄨󵄨 1−λ 1−λ =( ) 󵄨󵄨 󵄨󵄨 |h| |dz|, 2q 4 q 󵄨󵄨 dw 󵄨󵄨 N |W (f0 , . . . , fk )|1+ N ∏j=1 (∏k−1 s=0 |φjs |)

that is,

5.3 The Gauss curvature estimate q

� 145

∏j=1 |aj0 f0 + ⋅ ⋅ ⋅ + ajk fk |ω(j) χ |dw| =( ) |h|. 2q 4 q |dz| N |W (f0 , . . . , fk )|1+ N ∏j=1 (∏k−1 s=0 |φjs |) m

̃ 2m |ω|2 . Then Denote by l(Γa0 ) the length of the curve Γa0 with respect to the metric ‖G‖ 2 using |φjs | ≤ |Fs (Hj )| and noticing that ‖Fk ‖ = |W (f0 , . . . , fk )| and F(Hj ) = aj0 f0 +⋅ ⋅ ⋅+ajk fk , we get ̃ m |ω| = ∫ Ψ∗ (‖G‖ ̃ m |ω|) = ∫ ‖F‖m |h ∘ Ψ| l(Γa0 ) = ∫ ‖G‖ Γa0

≤ ∫(

La0

‖F‖χ ‖Fk ‖

La0

2q 1+ N

4

q

q

N ∏j=1 (∏k−1 s=0 |Fs (Hj )|)

∏j=1 |F(Hj )|ω(j)

La0

|dz| |dw| |dw|

m χ

) |dw|.

Applying Lemma 4.4.9, we get R

l(Γa0 ) ≤ C ∫( 0

λ

2R ) |dw| < ∞ R2 − |w|2

̃ 2m |ω|2 , since 0 < λ < 1. This contradicts the completeness assumption of the metric ‖G‖ and so we complete the proof of the theorem.

5.3 The Gauss curvature estimate In this subsection, we study the minimal surfaces without the completeness condition. In particular, we prove Theorem 5.3.2, which generalizes Theorem 5.2.1. To do so, we first prove a weaker result. Theorem 5.3.1 ([13]). Let M be an open Riemann surface, and let G : M → ℙn (ℂ) be a nonconstant holomorphic map. Consider the conformal metric on M given by ̃ 2m |ω|2 , ds2 = ‖G‖ where G̃ is a reduced representation of G, ω is a holomorphic 1-form on M, and m ∈ ℕ. Assume that G omits a neighborhood U of a hyperplane in ℙn (ℂ). Then there exists a constant C, which only depends on U (and not on G and M), such that 󵄨󵄨 󵄨1 󵄨󵄨K(p)󵄨󵄨󵄨 2 d(p) ≤ C,

(5.6)

where K(p) is the Gauss curvature of the surface at p, and d(p) is the geodesic distance from p to the boundary of M.

146 � 5 Gauss maps of minimal surfaces in ℝm If we assume that M is complete with respect to the induced metric, then d(p) = ∞ for any p ∈ M. Hence K ≡ 0, so the minimal must lie on a plane. We derive an expression for the Gauss curvature K of M with the conformal metric 2 ̃ 2m |ω|2 . Since ω is a holomorphic 1-form, from (3.6) we have that the Gauss ds = ‖G‖ curvature of ds2 is K = −m

̃ ∑0≤i 0 such that |gn | ≥ϵ ̃ ‖G‖ on Δ, where G̃ = (g0 , . . . , gn ). Let ψj :=

gj . gn

(5.10)

Then this implies that

n−1

1 󵄨 󵄨2 ∑ 󵄨󵄨󵄨ψj (z)󵄨󵄨󵄨 ≤ 2 − 1. ϵ j=0

(5.11)

Denote 󵄨 󵄨 Cj := 󵄨󵄨󵄨ψj (0)󵄨󵄨󵄨,

󵄨 󵄨 Dj := 󵄨󵄨󵄨ψ′j (0)󵄨󵄨󵄨,

󵄨 󵄨 Mj := sup󵄨󵄨󵄨ψj (z)󵄨󵄨󵄨. |z| 0, and let h be a function on Sn (r) such that hσn is integrable over Sn (r). Let p(w) = √r 2 − |w|2 . Then

162 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation 1 r 2n−2

∫ hσn = Sn (r)

h(w, ζ )σ1 (ζ ))ρn−1 (w).

∫ ( ∫ Bn−1 [r] S1 (p(w))

Proof. Define E = {(z1 , . . . , zn ) ∈ Sn (r) | 0 ≤ zn ∈ ℝ} and F = Sn (r) − E. Then E is a closed subset of Sn (r) of zero (2n − 1)-dimensional Hausdorff measure. A bijective map g : ℝ(0, 2π) × Bn−1 (r) → F of class C ∞ is defined by g(ϕ, w) = (w, p(w)eiϕ ) for all ϕ ∈ ℝ(0, 2π) and w ∈ Bn−1 (r). Let ϕ be the variable in ℝ(0, 2π), and let w1 , . . . , wn−1 be the complex variables on ℂn−1 with wj = xj + yj √−1. The partial derivatives gϕ , gx1 , gy1 , . . . , gxn−1 , gyn−1 are pointwise linearly independent over ℝ, and g is perpendicular to these derivatives. Here g(ϕ, w) points in the direction of Sn−1 (r) at g(ϕ, w). Since det(g, gϕ , gx1 , gy1 , . . . , gxn−1 , gyn−1 ) = r 2 > 0, the map g is an orientation-preserving diffeomorphism. We will compute g ∗ (σn ). For z = (z1 , . . . , zn ) ∈ ℂn and w = (w1 , . . . , wn−1 ) ∈ ℂn−1 , we have g ∗ (d c |z|2 ) = d c |w|2 + (1/2π)(r 2 − |w|2 )dϕ,

g ∗ (dd c |z|2 ) = dd c |w|2 − (1/2π)d|w|2 ∧ dϕ, n−1

g ∗ (dd c |z|2 ) g ∗ (d c |z|2 ∧ (dd c |z|2 )

n−1

= (dd c |w|2 )

n−1

−

n−1 n−2 d|w|2 ∧ dϕ ∧ (dd c |w|2 ) , 2π

) = (1/2π)(r 2 − |w|2 )dϕ ∧ (dd c |w|2 ) +

n−1

n−1 n−2 dϕ ∧ d|w|2 ∧ d c |w|2 ∧ (dd c |w|2 ) . 2π

It is easy to check that |w|2 (dd c |w|2 )n−1 = (n − 1)d|w|2 ∧ d c |w|2 ∧ (dd c |w|2 )n−2 , and hence n−1

g ∗ (d c |z| ∧ (dd c |z|2 )

) = (1/2π)r 2 dϕ ∧ (dd c |w|2 )

n−1

.

Now |g|2 = r 2 and (dd c |w|2 )n−1 = ρn−1 , and we obtain g ∗ (σn ) = g ∗ (d c log |z|2 ∧ (dd c log |z|2 ) = (1/2π)r

2−2n

dϕ ∧ ρn−1 .

n−1

Fubini’s theorem implies that ∫ hσn = r Sn (r)

2−2n

2π

∫ Bn−1 [r]

1 ∫ h(w, p(w)eiϕ )dϕ ∧ ρn−1 (w). 2π 0

n−1

) = g ∗ (r −2n d c |z|2 ∧ (dd c |z|2 )

)

6.1 Meromorphic functions on B(R0 ) ⊂ ℂn

�

163

Lemma 6.1.2. Let ϕ be a nonconstant meromorphic function on B(R0 ) with 0 < R0 ≤ ∞, and let a ∈ ℂ ∪ {∞}. Then, for 0 < r < R0 , 1 r 2n−2

∫ nϕ[w] (√r 2 − |w|2 , a)ρn−1 (w) ≤ nϕ (r, a), Bn−1 [r]

where ϕ[w] = ϕ(w, z) for w ∈ ℂn−1 and z ∈ ℂ. Proof. For each j ∈ ℕ[1, n], define πj : ℂn → ℂn−1 by πj (z1 , . . . , zn ) = (z1 , . . . , zj−1 , zj+1 , . . . , zn ). Then n

∗ n−1 ∗ υn−1 n = ∑ πj (υn−1 ) ≥ πn (ρn−1 ). j=1

Define A = Bn (r) ∩ supp νϕa . If A is empty, then the lemma is trivial. Assume that A is not empty. Then A is a pure (n − 1)-dimensional subset of Bn (r). Denote π = πn |A . Then π(A) ⊂ Bn−1 (r). The set E = {z ∈ A | rankz π < n−1} is analytic in Bn (r). Let E0 be the union of all (n−1)-dimensional branches of E, and let E1 be the union of all other branches of E. Then E0 and E1 are analytic in Bn (r) with dim E1 ≤ n − 2 and dim E0 = n − 1 if E0 ≠ 0. Also, E0′ = π(E0 ) is analytic in Bn−1 (r) with E0 = πn−1 (E0′ ) ∩ Bn (r). Therefore π ∗ (ρn−1 ) = 0 on E0 . The complement A0 = A − E0 is open in A, and A0 ∩ E1 is thin analytic in A0 if A0 ≠ 0. Then A1 = A0 − E1 = A − E is open in A. There exists a thin analytic subset E2 of A1 such that π is locally biholomorphic on A2 = A1 − E2 . Then F = π(E0 ∪ E1 ∪ E2 ) has a zero measure in ℂn−1 . Also, B2 = π(A2 ) is open in ℂn−1 . Here B = π(A) = B2 ∪ F and B3 = B2 − F differ from B2 by sets of zero measure. The intersection Sn (r) ∩ supp νϕa has a (2n − 2)-dimensional Hasudorff measure of zero. If w ∈ Bn−1 (r), then π −1 (w) = ({w} × B1 (√r 2 − |w|2 )) ∩ A, and for almost all w ∈ Bn−1 (r), we have π −1 (w) = ({w} × B1 [√r 2 − |w|2 ]) ∩ supp νϕa . For all w ∈ Bn−1 (r) − B, we have π −1 (w) = 0, and hence nf[w] (√r 2 − |w|2 , a) = 0 for almost all w ∈ Bn−1 (r) − B. If w ∈ B3 , then π −1 (w) = ({w} × B1 (√r 2 − |w|2 )) ∩ A. Take P = (b, c) ∈ A2 with b = π(P) ∈ Bn−1 (r) and c ∈ ℂ. There exist open connected neighborhoods V of b in Bn−1 (r), W of c, and U = V × W of P in Bn (r) with U ∩ A2 = U ∩ A

164 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation such that π : U ∩ A → V is biholomorphic. Let λ be the inverse map. Then there exists a holomorphic function h : V → W such that λ(w) = (w, h(w)) for all w ∈ V . The multiplicity νϕa (w, z) = q is constant for all (w, z) ∈ U ∩ A. There exists a holomorphic function H : U → ℂ − {0} such that q

ϕ(w, z) = a + (z − h(w)) H(w, z) for all (w, z) ∈ U. Therefore νϕa (w, z) = q = νϕa [w] (z) for all (w, z) ∈ U ∩ A. All together, we obtain a ∗ a ∗ r 2n−2 nϕ (r, a) = ∫ νϕa υn−1 n ≥ ∫ νϕ π (ρn−1 ) = ∫ νϕ π (ρn−1 ) A

A

A0

∗

= ∫ νπ (ρn−1 ) = ∫ A2

= ∫( B3

∑

B2 (w,z)∈A2

∑

νϕa (w, z)ρn−1 (w)

νϕa [w] (z))ρn−1 (w)

z∈B1 (√r 2 −|w|2 )

= ∫ nϕ[w] (√r 2 − |w|2 , a)ρn−1 (w) B3

= ∫ nϕ[w] (√r 2 − |w|2 , a)ρn−1 (w) B

= ∫ nϕ[w] (√r 2 − |w|2 , a)ρn−1 (w). Bn−1 (r)

Theorem 6.1.3. Let ϕ be nonzero a meromorphic function on B(R0 ) ⊂ ℂn , 0 < R0 ≤ ∞, let α = (α1 , . . . , αn ) ≠ (0, . . . , 0), 0 < r0 < R0 , and let p, p′ be positive numbers such that 0 < p|α| < p′ < 1. Then, for r0 < r < R < R0 , p 󵄨󵄨 󵄨󵄨p Dα ϕ R2n−1 󵄨 󵄨 )(z)󵄨󵄨󵄨 σn (z) ≤ K( Tϕ (R)) , ∫ 󵄨󵄨󵄨zα ( 󵄨󵄨 󵄨󵄨 ϕ R−r ′

S(r)

α

α

where K is a constant not depending on r and R, and zα := z1 1 ⋅ ⋅ ⋅ znn for z = (z1 , . . . , zn ) and α = (α1 , . . . , αn ). Before proving this, we give the following corollary to Theorem 6.1.3, which is essentially the same as the lemma of the logarithmic derivative in several complex variables given by Vitter [86].

6.1 Meromorphic functions on B(R0 ) ⊂ ℂn

�

165

Corollary 6.1.4. Let α = (α1 , . . . , αn ) ≠ (0, . . . , 0), 0 < r0 < R0 . Then, for r0 < r < R < R0 , R2n−1 󵄨 󵄨 T (R)). ∫ log+ 󵄨󵄨󵄨(Dα ϕ/ϕ)(z)󵄨󵄨󵄨σn (z) ≤ K log+ ( R−r ϕ

S(r)

Proof of Theorem 6.1.3. First, write ϕ = ϕ(η, ζ ) with ζ ∈ ℂ. From (4.12) we have, with p̃ = p/p′ , ∫ |ζ |=√r 2 −|η|2

≤(

󵄨󵄨 𝜕ϕ 󵄨󵄨p̃ 󵄨󵄨 󵄨 󵄨󵄨ζ ( /ϕ)(η, ζ )󵄨󵄨󵄨 σ1 (ζ ) 󵄨󵄨 󵄨󵄨 𝜕ζ

ρ ρ−r

p̃

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨log󵄨󵄨ϕ(η, ζ )󵄨󵄨󵄨󵄨󵄨󵄨σ1 (ζ ))

∫ |ζ |=√r 2 −η2

+ K(nϕ (√ρ2 − |η|2 , 0) + nϕ (√ρ2 − |η|2 , ∞)).

(6.3)

We prove Theorem 6.1.3 by induction on |α|. We first consider the case where |α| = 1. Without loss of generality, we assume that Dα = Dn . Set ρ = (r + R)/2. Since each pole of Dn ϕ/ϕ is of order ≤ 1, |zn (Dn ϕ/ϕ)(z)|p̃ is integrable on S(r). By Lemma 6.1.1, Lemma 6.1.2, (6.3), and the Hölder inequality we have 󵄨 󵄨p̃ ∫ 󵄨󵄨󵄨zn (Dn ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z)

S(r)

=

≤

1

r 2n−2

r

∫ υn−1 n−1 (η) |η|≤r

󵄨󵄨 󵄨p̃ 󵄨󵄨ζ (Dn ϕ/ϕ)(η, ζ )󵄨󵄨󵄨 σ1 (ζ )

∫ |ζ |=√r 2 −η2

1−p̃

p̃

1

( 2n−2

ρ ) ( ∫ υn−1 n−1 (η)) ρ−r |η|≤r

× ( ∫ υn−1 n−1 (η) |η|≤r

+

K

r 2n−2

∫

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨log󵄨󵄨ϕ(η, ζ )󵄨󵄨󵄨󵄨󵄨󵄨σ1 (ζ ))

|ζ |=√ρ2 −η2

∫ {nϕ|η (√ρ2 − |η|2 ), 0) + nϕ|η (√ρ2 − |η|2 ), ∞)}υn−1 n−1 (η) |η|≤r p̃

≤(

2n−2

ρ ρ 󵄨 󵄨 󵄨󵄨 ∫ 󵄨󵄨󵄨log󵄨󵄨󵄨ϕ(η, ζ )󵄨󵄨󵄨󵄨󵄨󵄨σn ) + K( ) ρ−r r S(ρ)

Therefore

p̃

{nϕ|η (ρ, 0) + nϕ|η (ρ, ∞)}.

166 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation

󵄨p̃ 󵄨 󵄨p 󵄨 ∫ 󵄨󵄨󵄨zn (Dn ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z) ≤ ( ∫ 󵄨󵄨󵄨zn (Dn ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z))

p′

S(r)

S(r)

̃ ′ pp

ρ 󵄨 󵄨 ≤( ∫ 󵄨󵄨󵄨log |ϕ|󵄨󵄨󵄨σn ) ρ−r S(ρ)

(2n−2)p′

ρ + K( ) r

(nϕ|η (ρ, 0)p + nϕ|η (ρ, ∞)p ). ′

′

By the first main theorem 󵄨 󵄨 ∫ 󵄨󵄨󵄨log |ϕ|󵄨󵄨󵄨σn ≤ 2Tϕ (r) + K,

S(r)

and by (4.13) with a = 0 and ∞ nϕ (ρ, a) ≤

2R (T (R) + K), R−r ϕ

where K stands for a constant, which may be different each time. Thus 󵄨 󵄨p ∫ 󵄨󵄨󵄨zn (Dn ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z)

S(r)

p

2R 4R2n−1 󵄨 󵄨 ̃ (T (R) + K)) ≤( ∫ 󵄨󵄨󵄨log |ϕ|󵄨󵄨󵄨σn ) + K( R−r R−r ϕ

p′

S(ρ)

≤ K(

p′

R2n−1 T (R)) . R−r ϕ

(6.4)

This proves the theorem for the case Dα = Dn . We now consider the general case. Assume ′ that the theorem holds for the case |α| ≤ κ. Then Dα ϕ/ϕ = (Di ϕ/ϕ)(Dα (Di ϕ)/Di ϕ), zα = ′ zi zα , and |α|p = (|α′ | + 1)p < p′ < 1. Set p1 := 1/(|α′ | + 1) and p2 := |α′ |/(|α′ | + 1). By the Hölder inequality, the induction hypothesis, and (6.4) we have 󵄨 󵄨p ∫ 󵄨󵄨󵄨zα (Dα ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z)

S(r)

p1

p2

′ 󵄨 󵄨p/p 󵄨 ′ 󵄨p/p ≤ ( ∫ 󵄨󵄨󵄨zi (Di ϕ/ϕ)(z)󵄨󵄨󵄨 1 σn (z)) ( ∫ 󵄨󵄨󵄨zα (Dα (Di ϕ)/Di ϕ)(z)󵄨󵄨󵄨 2 σn (z))

S(r)

S(r)

2n−1

p′ p1

R T (R)) ≤ K( R−r ϕ Notice that

2n−1

R ( T (R)) R − r Di ϕ

p′ p2

.

(6.5)

6.2 Nevanlinna theory on B(R0 ) ⊂ ℂn

� 167

TDi ϕ (r) ≤ TDi ϕ/ϕ (r) + Tϕ (r) + K ≤ ∫ log+ |Di ϕ/ϕ|σn + NDi ϕ/ϕ (r, ∞) + Tϕ (r) + K. S(r)

Since NDi ϕ/ϕ (r, ∞) ≤ Nϕ (r, ∞) + Nϕ (r, 0) ≤ 2Tϕ (r) + K and R2n−1 󵄨 󵄨 T (R)) ∫ log+ 󵄨󵄨󵄨(Di ϕ/ϕ)(z)󵄨󵄨󵄨σn ≤ K log+ ( R−r ϕ

S(r)

by (6.4), we have TDi ϕ (r) ≤ 3Tϕ (r) + K log+ (

R2n−1 T (R)) for i = 1, 2, . . . , n. R−r ϕ

(6.6)

For any ϵ > 0, there exists a positive constant Kϵ such that log+ (

ϵ

R2n−1 R2n−1 Tϕ (R)) ≤ Kϵ ( T (R)) . R−r R−r ϕ

(6.7)

Thus, for slightly smaller p′ if necessary, by combining (6.5), (6.6), and (6.7) we can conclude p′

R2n−1 󵄨 󵄨p T (R)) . ∫ 󵄨󵄨󵄨zα (Dα ϕ/ϕ)(z)󵄨󵄨󵄨 σn (z) ≤ K( R−r ϕ

S(r)

This completes the proof of the theorem.

6.2 Nevanlinna theory on B(R0 ) ⊂ ℂn Let f : B(R0 ) ⊂ ℂn → ℙN (ℂ) be a meromorphic map with 0 < R0 ≤ ∞. We take holomorphic functions f0 , f1 , . . . , fN such that If := {z ∈ B(R0 ) | f0 (z) = ⋅ ⋅ ⋅ = fN (z) = 0} is of dimension at most n − 2 and f (z) = [f0 (z) : ⋅ ⋅ ⋅ : fN (z)] on B(R0 ) − If in terms of homogeneous coordinates [w0 : ⋅ ⋅ ⋅ : wN ] on ℙN (ℂ). We call such a representation f = (f0 , . . . , fN ) a reduced representation of f . The pullback of the normalized Fubini– Study metric form Ω on ℙN (ℂ) by f is given by Ωf = dd c log ‖f ‖2 , where ‖f ‖ := max0≤j≤N |fj |. The characteristic function of f is defined as

168 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation r

Tf (r) = ∫ r0

dt ∫ Ωf ∧ υn−1 n t 2n−1 B(t)

for 0 < r0 < r < R0 , which is independent of r0 up to a bounded term. Let H be a hyperplane defined by a, that is, H = {[w0 : ⋅ ⋅ ⋅ : wN ] | a0 w0 + ⋅ ⋅ ⋅ + aN wN = 0} ⊂ ℙN (ℂ), where a = (a0 , . . . , aN ). The proximity function mf (r, H) for 0 < r < R0 is defined by mf (r, H) = ∫ λH (f )σn ,

(6.8)

S(r)

where the Weil function λH (f ) of f with respect to the hyperplane H is λH (f )(z) = log

‖f(z)‖‖a‖ . | < f(z), a > |

The first main theorem states (see [83], pp. 251–255) that Tf (r) = mf (r, H) + Nf (r, H) + O(1) for 0 < r < R0 . Next, we prove the second main theorem. We say a map f : B(R0 ) ⊂ ℂn → ℙN (ℂ) is linearly nondegenerate if f (B(R0 )) ⊄ H for every hyperplane H in ℙN (ℂ) or, equivalently, f0 , . . . , fN are linearly independent, where f = (f0 , . . . , fN ) is a reduced representation of f . Recall that for a holomorphic function h on an open domain G ⊂ ℂn and a set α = (α1 , . . . , αn ) of integers αi ≥ 0, we denote |α| := α1 + ⋅ ⋅ ⋅ + αn and α α Dα h := D1 1 ⋅ ⋅ ⋅ Dnn h, where Di h = (𝜕/𝜕zi )h for i = 1, . . . , n. Take an arbitrary set αi = (αi1 , . . . , αin ), 1 ≤ i ≤ N +1, of nonnegative integers. We define the generalized Wronskian of f as i

Wα1 ⋅⋅⋅αN+1 (f ) = det(Dα f : 1 ≤ i ≤ N + 1), where f = (f0 , . . . , fN ) is a reduced representation of f , and Dα f = (Dα f0 , . . . , Dα fN ). We prove the following theorem. Theorem 6.2.1. Let f : B(R0 ) → ℙN (ℂ) be a linearly nondegenerate meromorphic map. Then there exist αj = (αj1 , . . . , αjn ) with integers αji ≥ 0, |αj | ≤ N for 1 ≤ j ≤ N + 1, and |α1 | + ⋅ ⋅ ⋅ + |αN+1 | ≤ N(N + 1)/2, such that Wα1 ⋅⋅⋅αN+1 (f ) ≢ 0. To prove Theorem 6.2.1, we denote by ℳp the field of all germs of meromorphic functions at a point p ∈ B(R0 ) and by ℱ κ the ℳp -submodule of ℳN+1 generated by p

κ {Dα f : |α| ≤ κ}, where Dα f = (Dα f0 , . . . , Dα fN ) ∈ ℳN+1 p . Set ℱ = ⋃κ ℱ .

6.2 Nevanlinna theory on B(R0 ) ⊂ ℂn

�

169

Proposition 6.2.2. The set ℱ κ does not depend on the choice of holomorphic local coordinates (z1 , . . . , zn ) and reduced representation f = (f0 , . . . , fN ) of f . Proof. We will show this by induction on κ. For the case κ = 0, we have nothing to prove. Assume that the statement holds for |α| ≤ κ. We consider another system of holomorphic local coordinates u = (u1 , . . . , un ). We denote Di,u =

𝜕 , 𝜕ui

Dα,u =

𝜕|α| α α 𝜕u1 1 ⋅ ⋅ ⋅ 𝜕unn

for α = (α1 , . . . , αn ). For an arbitrary α with |α| = κ + 1, we write Dα = Di Dα , where 1 ≤ i ≤ n and |α′ | = κ. By the induction hypothesis we can write ′

Dα ,u f = ∑ hβ Dβ f ′

|β|≤κ

with some hβ ∈ ℳp . Then n

Dα,u f = Di,u (Dα ,u f) = ∑ Di,u hβ Dβ f + ∑ hβ ∑ ′

|β|≤κ

|β|≤κ

j=1

𝜕zj 𝜕ui

Dj Dβ f ∈ ℱ κ+1 .

This proves that the set ℱ κ+1 does not depend on a particular choice of a system of holomorphic local coordinates. Next, we consider another reduced representation f ̃ = (f0̃ , . . . , fÑ ) of f . Then hi := ′ fĩ /fi , i = 0, . . . , N, are nowhere zero holomorphic functions. Let Dα = Di Dα for |α′ | = κ. By the induction hypothesis we can write Dα f ̃ = ∑ gβ Dβ f ′

|β|≤κ

with some gβ ∈ ℳp . Therefore Dα f ̃ = Di (Dα f ̃) = ∑ Di gβ Dβ f + ∑ gβ Di Dβ f ∈ ℱ κ+1 . ′

|β|≤κ

|β|≤κ

Hence the set ℱ κ+1 does not depend on a particular choice of reduced representations. The proposition is thus proved by induction. Proposition 6.2.3. The map f : B(R0 ) ⊂ ℂn → ℙN (ℂ) is linearly nondegenerate if and only if ℱ = ℳN+1 for some point p or, equivalently, l(κ0 ) = N + 1 for some κ0 , where p l(κ) = dimℳp ℱ κ . Proof. Assume that f is degenerate. Then a0 f0 + ⋅ ⋅ ⋅ + aN fN ≡ 0

170 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation for some (a0 , . . . , an ) ≠ (0, . . . , 0). Therefore a0 Dα f0 + ⋅ ⋅ ⋅ + aN Dα fN ≡ 0 for all α. Therefore l(κ) = rankℳp {Dα f : |α| ≤ κ} < N + 1 for every κ = 1, 2, . . . . Conversely, assume that maxκ l(κ) < N +1. Then there exists (ϕ0 , . . . , ϕN ) ≠ (0, . . . , 0) in ℳN+1 such that p ϕ0 Dα f0 + ⋅ ⋅ ⋅ + ϕN Dα fN ≡ 0 for all α. Take a point q ∈ M sufficiently near p such that ϕ0 , . . . , ϕN are holomorphic in a neighborhood of q and (ϕ0 (q), . . . , ϕN (q)) ≠ (0, . . . , 0). Set Ψ(z) := ϕ0 (q)f0 (z) + ⋅ ⋅ ⋅ + ϕN (q)fN (z) on U. Then (Dα Ψ)(q) := ϕ0 (q)Dα f0 (q) + ⋅ ⋅ ⋅ + ϕN (q)Dα fN (q) = 0 for all α. Hence Ψ ≡ 0 is constant. This shows that f is degenerate. Lemma 6.2.4. If ℱ κ = ℱ κ+1 for a positive integer κ, then ℱ = ℱ κ . Proof. We show that ℱ κ ⊂ ℱ κ for every κ′ (> κ) by induction. There is nothing to prove ′ in the case κ′ = κ + 1. Assume that ℱ κ ⊂ ℱ κ for κ′ > κ. Take α with |α| = κ′ + 1 and write ′ Dα = Di Dα , where 1 ≤ i ≤ n and |α′ | = κ. By the induction hypothesis we can write ′

Dα f = ∑ ϕβ Dβ f ′

|β|≤κ

with some ϕβ ∈ ℳp . Then Dα f = Di (Dα ,u f) = ∑ Di ϕβ Dβ f + ∑ ϕβ Dj Dβ f ∈ ℱ κ+1 = ℱ κ . ′

|β|≤κ

|β|≤κ

Thus ℱ κ +1 ⊂ ℱ κ , and the lemma is proved by induction. ′

Proof of Theorem 6.2.1. By Proposition 6.2.3 there is at least one κ with l(κ) = N + 1. Set κ0 = min{κ : l(κ) = N + 1}. We claim that κ0 ≤ N. Indeed, Lemma 6.2.4 implies that 1 = l(0) < l(1) < ⋅ ⋅ ⋅ < l(κ0 ) = N + 1. Therefore

6.2 Nevanlinna theory on B(R0 ) ⊂ ℂn

�

171

κ0

N + 1 = l(κ0 ) = ∑ (l(κ) − l(κ − 1)) + l(0) ≥ κ0 + 1. κ=1

This gives κ0 ≤ N. Thus our claim is proved. Next, note that Wα1 ⋅⋅⋅αN+1 (f ) ≢ 0 if and only if Dα1 f, . . . , DαN+1 f is a basis of ℱ as an ℳp -module. We choose a basis of ℱ according to the following filtration by using Lemma 6.2.4 and the fact that l(κ0 ) = N + 1: 1

1

κ

ℱ ⊂ ℱ ⊂ ⋅ ⋅ ⋅ ⊂ ℱ 0,

that is, {Dα1 f, . . . , DαN+1 f} is a basis of ℱ such that {Dα1 f, . . . , Dαl(κ) f} is a basis of ℱ κ for κ = 1, 2, . . . , κ0 . Such a basis is called an admissible basis of ℱ . With this choice, we have κ0

󵄨󵄨 1 󵄨󵄨 󵄨 N+1 󵄨 󵄨󵄨α 󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨α 󵄨󵄨󵄨 ≤ ∑ κ(l(κ) − l(κ − 1)) κ=1 κ0

= ∑ κ(l(κ) − l(κ − 1)) κ=1

= κ0 l(κ0 ) − (l(0) + ⋅ ⋅ ⋅ + l(κ0 − 1)) ≤ κ0 (N + 1) − (1 + ⋅ ⋅ ⋅ + κ0 ) =

N(N + 1) − (N − κ0 )2 − (N − κ0 ) N(N + 1) ≤ . 2 2

From now on, we fix αj , 1 ≤ j ≤ N + 1, obtained from Theorem 6.2.1, so that Wα1 ⋅⋅⋅αN+1 (f ) ≢ 0, |αj | ≤ N for 1 ≤ j ≤ N + 1, and |α1 | + ⋅ ⋅ ⋅ + |αN+1 | ≤ N(N + 1)/2. Let Hj = {aj0 w0 + ⋅ ⋅ ⋅ + ajN wN = 0}, 1 ≤ j ≤ q, be hyperplanes in ℙN (ℂ) located in general position. Write Fj := aj0 f0 + ⋅ ⋅ ⋅ + ajN fN for 1 ≤ j ≤ q. Theorem 6.2.5. In the above situation, set l0 = |α1 | + ⋅ ⋅ ⋅ + |αN+1 | and take t and p′ such that 0 < tl0 < p′ < 1. Then, for 0 < r0 < R0 , there exists a positive constant K such that for r0 < r < R < R0 , p 󵄨󵄨 1 󵄨t N+1 W 1 R2n−1 󵄨 t(q−N−1) α ⋅⋅⋅αN+1 (f ) 󵄨󵄨󵄨 Tf (R)) . σn ≤ K( ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α 󵄨󵄨 ‖f ‖ 󵄨󵄨 F1 ⋅ ⋅ ⋅ Fq 󵄨󵄨 R−r ′

S(r)

Proof. Since H1 , . . . , Hq are in general position, similarly to Theorem 4.2.2, we have 󵄨󵄨 W 1 N+1 (f ) 󵄨󵄨 󵄨󵄨 Wα1 ⋅⋅⋅αN+1 (Fj , . . . , Fj ) 󵄨󵄨 󵄨󵄨 α ⋅⋅⋅α 󵄨󵄨 q−N−1 󵄨󵄨 1 N+1 󵄨󵄨 ≤ K( ∑ 󵄨󵄨 󵄨󵄨‖f ‖ 󵄨󵄨 󵄨󵄨) 󵄨󵄨 F1 ⋅ ⋅ ⋅ Fq 󵄨󵄨 󵄨󵄨 󵄨󵄨 F ⋅ ⋅ ⋅ F j1 jN+1 1≤j1 0. Since Tf (r) is continuous and increasing and we may assume that Tf (r) ≥ 1, we can apply Lemma 4.1.7 to show Tf (r +

R0 − r , r ) ≤ 2Tf (r) eTf (r) 0

outside a set E of r such that ∫E 1/(R0 − r)dr < ∞ in the case R0 < ∞ and Tf (r +

1 ) ≤ 2Tf (r) Tf (r)

outside a set E ′ of r such that ∫E′ dr < ∞ in the case R0 = ∞. Substituting R = r +

if R0 < ∞ and R = r + 1/Tf (r) if R0 = ∞ into (6.11) proves the theorem.

R0 −r eTf (r)

Corollary 6.2.7. In the situation of Theorem 6.2.6, if (i)

lim sup r→R0

Tf (r)

log(1/(R0 − r))

=∞

or (ii) R0 = ∞, then f ,⋆

∑ δN (Hj ) ≤ N + 1,

(6.12)

j

f ,⋆

where δN is the classical Nevanlinna defect defined as f ,⋆

δN (Hj ) = 1 − lim sup

Nf(N) (r, Hj )

r→R0

Tf (r)

.

6.3 The nonintegrated defect relation Let M be an n-dimensional complex Kähler manifold. Let f be a meromorphic map of M into ℙN (ℂ), let μ0 be a positive integer, and let D be a hypersurface in ℙN (ℂ) of degree d such that f (M) ⊄ D. We denote the intersection multiplicity of the image of f and D at f (p) by νf (D)(p) and the pullback of the normalized Fubini–Study metric form Ω on ℙN (ℂ) by Ωf . The nonintegrated defect of f with respect to D cut by μ0 is defined as

6.3 The nonintegrated defect relation

� 175

δμf 0 (D) := 1 − inf{η ≥ 0 : η satisfies condition (⋆)}. Here condition (⋆) means that there exists a bounded nonnegative continuous function h on M with zeros of order not less than min(νf (D), μ0 ) such that dηΩf +

√−1 𝜕𝜕̄ log h2 ≥ [min(νf (D), μ0 )], 2π

where d is the degree of D, and by [ν] we mean the (1, 1)-current associated with a divisor ν. Note that condition (⋆) also means that for each holomorphic function ϕ(≢ 0) on an open subset U of M with νϕ = min(νf (D), μ0 ) outside an analytic set of codimension ≥ 2, the function u := log(h2 ‖f ‖2dη /|ϕ|2 ) is continuous and plurisubharmonic on U, where ‖f ‖2 = |f0 |2 + ⋅ ⋅ ⋅ + |fN |2 , and f = [f0 : ⋅ ⋅ ⋅ : fN ] is a (local) reduced representation of f . Similarly to the classical Nevanlinna defect, we have the following properties. f f – 0 ≤ δμ0 (D) ≤ 1. To see that δμ0 (D) ≥ 0, take η = 1 and h = |Q(f )|/‖f ‖d , where Q is the homogeneous polynomial defining D; f – If f (M) ∩ D = 0, then by taking η = 0 and h = 1 we have that δμ0 (D) = 1;

–

f

If νf (D)(p) ≥ μ for all p ∈ f −1 (D) with some positive integer μ ≥ μ0 , then δμ0 (D) ≥ 1 − μ0 /μ by taking η = μ0 /μ and h = |Q(f )|/‖f ‖μ0 d/μ .

Let f : B(R0 ) ⊂ ℂn 󳨀→ ℙN (ℂ), 0 < R0 ≤ ∞, be a meromorphic map. Note that the f ,⋆ classical Nevanlinna defect δμ0 (D) of f with respect to D of degree d cut by μ0 is defined as δμf ,⋆ (D) = 1 − lim sup 0 r→R0

(μ0 )

Nf

(r, D)

dTf (r)

.

The relationship between the nonintegrated defect and the classical Nevanlinna defect is given as follows. Proposition 6.3.1. Let f : B(R0 ) ⊂ ℂn 󳨀→ ℙN (ℂ), 0 < R0 ≤ ∞, be a meromorphic map. If limr→R0 Tf (r) = ∞, then 0 ≤ δμf 0 (D) ≤ δμf ,⋆ (D) ≤ 1, 0 where δ⋆ is the classical Nevanlinna defect. f

Proof. Take η satisfying condition (⋆) in the definition of δμ0 (D). Then the function v := dη log‖f ‖+ log h − log |φ| is plurisubharmonic, where h is bounded, and φ is holomorphic on B(R0 ) with νφ = min(νf (D), μ0 ) outside an analytic set of codimension ≥ 2. Therefore

176 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation 0 ≤ ∫ vσn − ∫ vσn S(r)

S(r0 )

= dη ∫ log‖f ‖σn + ∫ log hσn − ∫ log |φ|σn + K S(r)

S(r)

≤ dηTf (r) −

(μ ) Nf 0 (r, D)

S(r)

+ K,

where K is a constant, because h is bounded from above. This implies that (μ0 )

Nf

(r, D)

dTf (r)

f ,⋆

≤η+ f ,⋆

K . Tf (r) f

As r → R0 , we obtain δμ0 (D) ≥ 1 − η. Hence δμ0 (D) ≥ δμ0 (D). Let M be an n-dimensional complex Kähler manifold. Let ω = the Kähler form of M. We define

√−1 2

∑ij hij dz ̄ i ∧ d z̄j be

Ric(ω) = dd c log(det(hij )), where d = 𝜕 + 𝜕̄ and d c = 4π−1 (𝜕̄ − 𝜕). Let f be a meromorphic map of M into ℙN (ℂ). We assume the following growth condition for f : there exists a nonzero bounded continuous real-valued function h on M such that √

ρΩf +

√−1 𝜕𝜕̄ log h2 ≥ Ric(ω) 2π

(6.13)

for some nonnegative constant ρ. Theorem 6.3.2 ([26]). Let M be an n-dimensional complete Kähler manifold, and let f : M → ℙN (ℂ) be a linearly nondegenerate meromorphic map. Assume that there exists a nonzero bounded continuous real-valued function h on M such that ρΩf +

√−1 𝜕𝜕̄ log h2 ≥ Ric(ω) 2π

(6.14)

for some nonnegative constant ρ. Assume that the universal covering of M is biholomorphic to a ball B(R0 ) ⊂ ℂn with 0 < R0 ≤ ∞. Let H1 , . . . , Hq be hyperplanes in ℙN (ℂ) located in general position. Then q

f

∑ δN (Hj ) ≤ N + 1 + ρN(N + 1). j=1

Proof. Since the universal covering of M is B(R0 ) ⊂ ℂn , by lifting f to the covering, we may assume that M = B(R0 ) ⊂ ℂn . In the case where R0 = ∞ and the case where

6.3 The nonintegrated defect relation �

lim sup

r󳨀→R0

Tf (r)

log 1/(R0 − r)

177

= ∞,

from Proposition 6.3.1 and Corollary 6.2.7 we have q

q

f

f ,⋆

∑ δN (Hj ) ≤ ∑ δN (Hj ) ≤ N + 1. j=1

j=1

Hence this case is proved. Now we consider the remaining case where R0 < ∞. Without loss of generality, we assume that R0 = 1 under the condition that lim sup

r󳨀→1

Tf (r)

log 1/(1 − r)

< ∞.

We assume that q

f

∑ δN (Hj ) > N + 1 + ρN(N + 1). j=1

By the definition of nonintegrated defect, there exist constants ηj ≥ 0 and continuous plurisubharmonic functions uj (≢ −∞) such that ρN(N + 1) q1 . By the assumption that the Fj are in general position we know that q1 ≤ N. Write γ :=

Wα1 ⋅⋅⋅αN+1 (F1 , . . . , FN+1 ) . F1 ⋅ ⋅ ⋅ FN+1

Then νψ∞ = νγ∞ since Fj does not vanish at a for j > N and Wα1 ⋅⋅⋅αN+1 (F1 , . . . , FN+1 ) = cWα1 ⋅⋅⋅αN+1 (f ). On the other hand, we can write γ = det(Dαi Fj /Fj : 1 ≤ i, j ≤ N + 1) = ∑ sgn(i1 ⋅ ⋅ ⋅ iN+1 )

DαN+1 FN+1 Dα1 F1 ⋅⋅⋅ . F1 FN+1

By (6.17) and the fact that |αj | ≤ N for 1 ≤ j ≤ N + 1 we get N+1

q

j=1

j=1

νγ∞ (a) ≤ ∑ min{νF0j (a), N} ≤ ∑ min{νF0j (a), N}. This proves the claim. Since uj − log |φj | is plurisubharmonic and νφ0 j = min(νf (Hj ), N), by the definition of v and (6.16), we see that v is plurisubharmonic on M = B(1). By the growth condition of f (see (6.14)) there exists a continuous plurisubharmonic function w ≢ −∞ on B(1) such that ew dV ≤‖f ‖2ρ vnn . Set

(6.18)

6.3 The nonintegrated defect relation

t=

� 179

2ρ q − N − 1 − (η1 + ⋅ ⋅ ⋅ + ηq )

(6.19)

and 1

χ := zα +⋅⋅⋅+α

N+1

Wα1 ...αN+1 (f ) . F1 ⋅ ⋅ ⋅ Fq

Define u := w + tv. Then u is plurisubharmonic and thus subharmonic on the Kähler manifold M. We then have eu dV ≤ ew+tv dV ≤ etv ‖f ‖2ρ υnn ≤ |χ|t ‖f ‖t(q−n−1) υnn . Therefore u

∫ e dV = ∫

eu υnn

B(1)

B(1)

1

≤ 2n ∫ r 2n−1 ( ∫ |χ|t ‖f ‖t(q−n−1) σn )dr, 0

S(r)

where we used the identity υnn = (dd c |z|2 )n = 2n|z|2n−1 σn ∧ d|z|. Note that t ⋅ so we take p′ such that t ⋅ N(N+1) < p′ < 1. Then 2 0 < (|α1 | + ⋅ ⋅ ⋅ + |αn+1 |)t ≤ t ⋅

N(N+1) 2

< 1,

N(N + 1) < p′ < 1. 2

Therefore by Theorem 6.2.5 ∫ |χ|t ‖f ‖t(q−n−1) σn ≤ K( S(r)

R2n−1 T (R)) R−r f

p′

for r0 < r < R < 1. According to Lemma 4.1.7, choose R := r + (1 − r)/eTf (r). Then Tf (R) ≤ 2Tf (r) outside a set such that ∫E 1/(1 − r)dr < ∞. Thus t

∫ |χ| ‖f ‖ S(r)

t(q−n−1)

p′

1 1 σn ≤ K(Tf (R)) ≤ K ) ′ (log 1−r (1 − r)p p′

for all r ∈ [r0 , 1)\E. Slightly varying the constant K, we may assume that this inequality holds for all r ∈ [r0 , 1) because of Proposition 4.1.8. From these facts we conclude that

180 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation 1

p′

t 2n−1 1 ) dt < ∞. ∫ e dV ≤ K ∫ ′ (log p 1 − t (1 − t) u

0

B(1)

On the other hand, by Theorem 3.3.9 (see Yau [90] or Karp [45]) we necessarily have ∫ eu dV = ∞, B(1)

because B(1) has infinite volume with respect to the given complete Kähler metric (cf. [45], Theorem B). This is a contradiction. So the proof of the theorem is completed. The rest of this chapter is to extend Theorem 6.3.2 to the case where f (M) intersects with hypersurfaces instead of hyperplanes; namely, we prove the following main theorem. Theorem 6.3.3 ([77]). Let M be an n-dimensional complete Kähler manifold, and let f : M → ℙN (ℂ) be an algebraically nondegenerate meromorphic map (i. e., its image is not contained in any proper subvariety of ℙN (ℂ)). Assume that the universal covering of M is biholomorphic to a ball B(R0 ) in ℂn with 0 < R0 ≤ ∞. Let D1 , . . . , Dq be hypersurfaces of degree dj in ℙN (ℂ) located in general position. Let d = l. c. m.{d1 , . . . , dq } (the least common multiple of {d1 , . . . , dq }). Assume that there exists a nonzero bounded continuous √ real-valued function h on M such that ρΩ + −1 𝜕𝜕̄ log h2 ≥ Ric(ω) for some nonnegative f

2π

constant ρ, where ω is the Kähler form of M. Then, for every ϵ > 0, q

f

∑ δl−1 (Dj ) ≤ N + 1 + ϵ + j=1

where l ≤ 2N number x.

2

ρl(l − 1) , d

+4N N 2N

e d (NI(ϵ−1 ))N , and I(x) := min{k ∈ ℕ : k > x} for a positive real

Proof. Since the universal covering of M is the unit ball B(R0 ) ⊂ ℂn , by lifting f to the covering we may assume that M = B(R0 ) ⊂ ℂm . Let D1 , . . . , Dq be hypersurfaces in ℙN (ℂ) of degrees d1 , . . . , dq located in general position. Let Qj , 1 ≤ j ≤ q, be the homogeneous d/d

polynomials defining Dj . Replacing Qj by Qj j if necessary, where d is the l. c. m. (the least common multiple) of dj , we can assume that Q1 , . . . , Qq have the same degree d. For m ∈ ℕ, let Vm be the space of homogeneous polynomials of N + 1 variables of degree m and fix an arbitrary basis ϕ1 , . . . , ϕl of Vm , where l = dim Vm . Set F = [ϕ1 (f ) : ⋅ ⋅ ⋅ : ϕl (f )]. Then F : B(R0 ) → ℙl−1 (ℂ) is linearly nondegenerate. By Theorem 6.2.1 there exist αj = (αj1 , . . . , αjl ) with integer αji ≥ 0 such that |αj | ≤ l − 1 for 1 ≤ j ≤ l, |α1 | + ⋅ ⋅ ⋅ + |αl | ≤ l(l − 1)/2, and the generalized Wronskian Wα1 ⋅⋅⋅αl (F) ≢ 0. Similarly to (4.44), for z ∈ B(R0 ), there are linear forms L1 , . . . , Lu of l-variables such that

6.3 The nonintegrated defect relation

� 181

‖f (z)‖dq△ ‖f (z)‖lm ‖F(z)‖l ′ ≤ K ≤ K , |Q1△ (f (z)) ⋅ ⋅ ⋅ Qq△ (f (z))| ∏lt=1 |Lt,z (F(z))| ∏lt=1 |Lt,z (F(z))| where △ is defined in (4.42), K > 0 is a constant, and {L1,z , . . . , Ll,z } ⊂ {L1 , . . . , Lu } are linearly independent. Hence ‖f (z)‖dq△−lm |Wα1 ⋅⋅⋅αl (F)(z)| |Q1△ (f (z)) ⋅ ⋅ ⋅ Qq△ (f (z))|

≤K

|Wα1 ⋅⋅⋅αl (F)(z)| , |L1 (F(z)) ⋅ ⋅ ⋅ Ll (F(z))|

(6.20)

where we just write Lj,z as Lj for simplicity. We first deal with the case R0 < ∞. We assume that R0 = 1. So we let f : B(1) 󳨀→ ℙN (ℂ) be an algebraically nondegenerate meromorphic map. We break the proof into the following two cases: the case lim sup

r󳨀→1

Tf (r)

log 1/(1 − r)

(n + 1) + ϵ + j=1

ρl(l − 1) . d

(6.22)

Then by the earlier discussion there exist constants ηj ≥ 0 and continuous plurisubharmonic functions ũ j (≢ −∞) such that eũ j |φj | ≤ ‖f ‖dηj for j = 1, . . . , q and q

q − ∑ ηj > n + 1 + ϵ + j=1

ρl(l − 1) , d

(6.23)

where φj is a nonzero holomorphic function with νφ0 j = min(νf (Dj ), l − 1). Let uj = ũ j + log |φj |. Then uj (≢ −∞), 1 ≤ j ≤ q, are continuous plurisubharmonic functions,

182 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation euj ≤ ‖f ‖dηj ,

(6.24)

and uj − log |φj | is plurisubharmonic, where φj is a nonzero holomorphic function with νφ0 j = min(νf (Dj ), l − 1). Let q 󵄨󵄨 1 l W 1 l (F) 󵄨󵄨󵄨 󵄨 v := log󵄨󵄨󵄨zα +⋅⋅⋅+α △ α ⋅⋅⋅α △ 󵄨󵄨󵄨 + △ ∑ uj . 󵄨󵄨 Q1 (f ) ⋅ ⋅ ⋅ Qq (f ) 󵄨󵄨 j=1

(6.25)

We now show that v is plurisubharmonic on M = B(1). To do so, set ψ=

Wα1 ⋅⋅⋅αl (F) . △ Q1 (f ) ⋅ ⋅ ⋅ Qq△ (f )

We claim that q

νψ∞ ≤ ∑ △ min{νQ0 j (f ) , l − 1}

(6.26)

j=1

outside an analytic set of codimension at least two. Indeed, let IF be the indeterminacy set of F, and take a ∈ B(1)\IF . For each a ∈ B(1)\IF , without loss of generality, we may assume that Qj (f ) vanishes at a for 1 ≤ j ≤ q1 and does not vanish at a for j > q1 . By the assumption that the Qj are in general position we know that q1 ≤ n. For {Q1 , . . . , Qn } ⊂ {Q1 , . . . , Qq }, consider the filtration Vm = W0 ⊃ ⋅ ⋅ ⋅ Wi ⊃ Wi′ ⊃ ⋅ ⋅ ⋅ defined in (4.41) associated with {Q1 , . . . , Qn }, and take a basis ψ1 , . . . , ψl of Vm according to this filtration. Then there are linearly independent linear forms L1 , . . . , Ll such that ψt (f ) = Lt (F), 1 ≤ t ≤ l. Denote by W := Wα1 ...αl (F) the generalized Wronskian of F. By the basic properties of generalized Wronskian W = Wα1 ...αl (F) = CWα1 ⋅⋅⋅αl (L1 (F), . . . , Ll (F)) = CWα1 ⋅⋅⋅αl (ψ1 (f ), ⋅ ⋅ ⋅ , ψl (f )), where C is some constant. Let ψ be an element of the basis {ψ1 , . . . , ψl }. As we discussed i i earlier, we may write ψ = Q11 ⋅ ⋅ ⋅ QnN η with η ∈ Vm−dσ(i) . Therefore i

i

ψ(f ) = (Q1 (f )) 1 ⋅ ⋅ ⋅ (QN (f )) N η(f ). Note that there are △i such ψ is our basis, where △i = dim(Wi /Wi′ ) with i′ > i are consecutive n-tuples with Wi′ ⊂ Wi . Assume that νQ0 j (f ) (a) ≥ l − 1 for 1 ≤ j ≤ q0 and i

νQ0 j (f ) (a) < l − 1 for q0 < j ≤ q1 . Since, by the above, W = C det(Dα (ψj (f )))1≤i,j≤l , using

(6.17) and noticing that there are △i such ψ is our basis and that |αj | ≤ l − 1 for 1 ≤ j ≤ l, we have q0

0 νW (a) ≥ ∑(∑ ij (νQ0 j (a) − (l − 1)))△i i

j=1

6.3 The nonintegrated defect relation q0

� 183

q0

= ∑(∑ ij △i )(νQ0 j (a) − (l − 1)) = △ ∑(νQ0 j (a) − (l − 1)). j=1

j=1

i

On the other hand, q

N

q0

q1

j=1

j=1

j=1

j=q0

∑ νQ0 j (f ) (a) = ∑ νQ0 j (f ) (a) = ∑ νQ0 j (f ) (a) + ∑ νQ0 j (f ) (a). q

Hence νψ∞ (a) ≤ ∑j=0 △ min{νQ0 j (f ) (a), l − 1}. This proves (6.26). By the definition of v

(see (6.25)), using (6.26) and the fact that uj − log |φj | is plurisubharmonic and νφ0 j =

min(νf (Dj ), l − 1), we see that v is plurisubharmonic on M = B(1). We now continue our proof. By the growth condition of f (see (6.21)) there exists a continuous plurisubharmonic function w ≢ −∞ on B(1) such that ew dV ≤‖f ‖2ρ υnn .

(6.27)

Set t=

2ρ qd △ −lm − △d(η1 + ⋅ ⋅ ⋅ + ηq )

(6.28)

and 1

χ := zα +⋅⋅⋅+α

l

Wα1 ⋅⋅⋅αl (F) . △ Q1 (f ) ⋅ ⋅ ⋅ Qq△ (f )

Define u := w + tv. Then u is plurisubharmonic and thus subharmonic on the Kähler manifold M. By Theorem 3.3.9 (or Karp [45]) we necessarily have ∫ eu dV = ∞, B(1)

because B(1) has infinite volume with respect to the given complete Kähler metric (cf. [45], Theorem B). Now by (6.24), (6.27), and (6.28) eu dV = ew+tv dV ≤ etv ‖f ‖2ρ υnn q

q

= |χ|t (∏ et△uj )‖f ‖2ρ υnn ≤ |χ|t (∏‖f ‖t△dηj )‖f ‖2ρ υnn j=1

= |χ|

t

j=1

q

2ρ+td△∑j=1 ηj n ‖f ‖ υn

= |χ|

t

‖f ‖t(dq△−lm) υnn .

184 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation The contradiction will appear if we show that ∫ eu dV < ∞. B(1)

By Lemma 4.2.13, have

lm △

≤ d(N + 1) + ϵ. Thus qd −

lm △

≥ d(q − (N + 1 + ϵ)). So, using (6.23), we

q

q

j=1

j=1

dq △ −lm − △ ∑ dηj ≥ d △ (q − (N + 1 + ϵ)) − △ ∑ dηj > △ρl(l − 1). This implies that tl(l − 1)/2 < 1. Since |α1 | + ⋅ ⋅ ⋅ + |αl | ≤ l(l − 1)/2, we can choose p′ such that t(|α1 | + ⋅ ⋅ ⋅ + |αl |) ≤ tl(l − 1)/2 < p′ < 1. By the identity n

υnn = (dd c |z|2 ) = 2n|z|2n−1 σn ∧ d|z| we have ∫ eu dV ≤ ∫ |χ|t ‖f ‖t(dq△−lm) υnn B(1)

B(1) 1

≤ 2n ∫ r 2n−1 ( ∫ |χ|t ‖f ‖t(dq△−lm) σn )dr 0

S(r)

1

(dq△−lm) 󵄨󵄨 1 l W 1 󵄨󵄨󵄨t l (F)‖f ‖ 󵄨 󵄨 = 2n ∫ r 2n−1 ( ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α α ⋅⋅⋅α σ n 󵄨󵄨󵄨 )dr. 󵄨󵄨 Q1△ (f ) ⋅ ⋅ ⋅ Qq△ (f ) 󵄨 0

(6.29)

S(r)

On the other hand, by (6.20) |Wα1 ⋅⋅⋅αl (F)|‖f ‖(dq△−lm) |Q1△ (f ) ⋅ ⋅ ⋅ Qq△ (f )|

≤K ∑ ( L1 ,...,Ll

|Wα1 ⋅⋅⋅αl (F)| ), |L1 (F) ⋅ ⋅ ⋅ Ll (F)|

(6.30)

where the summation is taken over all possible linear forms L1 , . . . , Ll , which is a finite summation. By Theorem 6.2.5, for all L1 , . . . , Ll , t p 󵄨󵄨 1 l Wα1 ⋅⋅⋅αl (F) 󵄨󵄨󵄨󵄨 R2n−1 󵄨 TF (R)) . ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α 󵄨󵄨 σn ≤ K( 󵄨󵄨 L1 (F) ⋅ ⋅ ⋅ Ll (F) 󵄨󵄨 R−r ′

(6.31)

S(r)

Combining (6.30) and (6.31) thus gives t p 󵄨󵄨 1 l W 1 l (F) 󵄨󵄨󵄨 R2n−1 󵄨 TF (R)) ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α △ α ⋅⋅⋅α △ 󵄨󵄨󵄨 ‖f ‖t(dq△−lm) σn ≤ K( 󵄨󵄨 R−r Q1 (f ) ⋅ ⋅ ⋅ Qq (f ) 󵄨󵄨

′

S(r)

(6.32)

6.3 The nonintegrated defect relation

� 185

for r0 < r < R < 1, where, as we noted, we use the letter K to denote a constant depending only on f and D1 , . . . , Dq , even when it should be replaced by a new constant. According to Lemma 4.1.7, if we choose R := r + (1 − r)/eTF (r), then TF (R) ≤ 2TF (r) ≤ 2dTf (r) outside a set E such that ∫E 1/(1 − r)dr < ∞. If lim sup

r󳨀→1

Tf (r)

log 1/(1 − r)

< ∞,

then (6.32) becomes p t 󵄨󵄨 1 l W 1 l (F) 󵄨󵄨󵄨 K 1 󵄨 ) (log ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α △ α ⋅⋅⋅α △ 󵄨󵄨󵄨 ‖f ‖t(dq△−lm) σn ≤ ′ 󵄨󵄨 1−r Q1 (f ) ⋅ ⋅ ⋅ Qq (f ) 󵄨󵄨 (1 − r)p

′

(6.33)

S(r)

for all r ∈ [0, 1) outside a set E such that ∫E 1/(1 − r)dr < ∞. Slightly varying a constant K, we may assume that this inequality holds for all r ∈ [0, 1) because of Proposition 4.1.8. Therefore by (6.29) and (6.33) we have 1

p′

r 2n−1 1 ) dr < ∞, ∫ e dV ≤ K ∫ ′ (log p 1−r (1 − r) u

0

B(1)

since p′ < 1. This gives a contradiction and thus completes the proof for the first case. We now deal with the case where either R0 = ∞ or R0 < ∞ with lim sup

r󳨀→1

Tf (r)

log 1/(R0 − r)

= ∞.

Similarly to the proof of (6.32), using (6.20) and Theorem 6.2.5, we have p t 󵄨󵄨 1 l W 1 l (F) 󵄨󵄨󵄨 R2n−1 󵄨 TF (R)) ∫ 󵄨󵄨󵄨zα +⋅⋅⋅+α △ α ⋅⋅⋅α △ 󵄨󵄨󵄨 ‖f ‖t(dq△−lm) σn ≤ K( 󵄨󵄨 R−r Q1 (f ) ⋅ ⋅ ⋅ Qq (f ) 󵄨󵄨

′

(6.34)

S(r)

for r0 < r < R < R0 . Hence, by the concavity of the logarithm this inequality implies that 󵄨󵄨 W 1 l (F) 󵄨󵄨 l 󵄨 1 󵄨 󵄨 󵄨 t ∫ log󵄨󵄨󵄨zα +⋅⋅⋅+α 󵄨󵄨󵄨σn + t ∫ log󵄨󵄨󵄨 △ α ⋅⋅⋅α △ 󵄨󵄨󵄨σn 󵄨󵄨 Q (f ) ⋅ ⋅ ⋅ Q (f ) 󵄨󵄨 q 1 S(r)

S(r)

+ t(dq △ −lm) ∫ log‖f ‖σn ≤ p′ (log+

2n−1

S(r)

R + log+ TF (R)) + O(1) R−r

(6.35)

186 � 6 Nevanlinna theory on the open ball B(R0 ) ⊂ ℂn and the nonintegrated defect relation for r0 < R < R0 . However, by the Jensen formula (see Corollary 4.1.3) and (6.26) q 󵄨󵄨 W 1 l (F) 󵄨󵄨 󵄨 󵄨 ∫ log󵄨󵄨󵄨 △ α ⋅⋅⋅α △ 󵄨󵄨󵄨σn ≤ △ ∑ Nf[l−1] (r, Dj ). 󵄨󵄨 Q (f ) ⋅ ⋅ ⋅ Q (f ) 󵄨󵄨 q j=1 1

S(r)

Also TF (R) ≤ mTf (R) + O(1). Therefore (6.35) becomes q

(dq △ −lm)Tf (r) ≤ ∑ △Nf[l−1] (r, Dj ) j=1

+ K(log r + log+

R + log+ Tf (R)) + O(1). R−r

By Lemma 4.2.13 with m = 2d(N + 1)(Nd + N)(2N − 1)(I(ϵ−1 ) + 1) + Nd for any ϵ > 0 we have lm ≤ d(N + 1) + ϵ, △ and, moreover, l satisfies l ≤ 2N

2

+4N N 2N

e d (NI(ϵ−1 ))N . Hence q

(q − N − 1 − ϵ)Tf (r) ≤ ∑ d −1 Nf[l−1] (r, Dj ) j=1

+ K(log r + log+ In the case R0 = ∞, we take R = r +

1 . Tf (r)

R + log+ Tf (R)). R−r

By Lemma 4.1.7 we still have Tf (R) ≤ Tf (r) + 1

for all r outside a set E of finite measure. Thus we have q

(q − N − 1 − ϵ)Tf (r) ≤ ∑ d −1 Nf[l−1] (r, Dj ) + O(log Tf (r) + log r) j=1

for every r ∈ [0, ∞) excluding a set E ′ such that ∫E′ dt < ∞. Now consider the case where R0 < ∞ and lim sup r→R0

Tf (r)

log(1/(R0 − r))

= ∞.

Since Tf (r) is continuous and increasing, we may assume that Tf (r) ≥ 1. By Lemma 4.1.7 Tf (r +

R0 − r ) ≤ 2Tf (r) eTf (r)

6.3 The nonintegrated defect relation

outside a set E of r such that ∫E 1/(R0 − r)dr < ∞. Taking R = r + q

(q − N − 1 − ϵ)Tf (r) ≤ ∑ d −1 Nf[l−1] (r, Dj ) + K(log+ j=1

for every r ∈ [0, R0 ) excluding a set E such that ∫E q

1 dt R0 −t

R0 −r , eTf (r,r0 )

� 187

we obtain

1 + log+ Tf (r)) R0 − r

< ∞. In both cases, we get

f ,⋆

∑ δl−1 (Dj ) ≤ N + 1. j=1

Hence q

f

q

f ,⋆

∑ δl−1 (Hj ) ≤ ∑ δl−1 (Hj ) ≤ N + 1. j=1

j=1

Note that in the second part of the theorem, we indeed proved the following second main theorem for mappings intersecting hypersurfaces. Theorem. Let f : B(R0 ) ⊂ ℂm 󳨀→ ℙn (ℂ), 0 < R0 ≤ ∞, be an algebraically nondegenerate meromorphic map, and let D1 , . . . , Dq be hypersurfaces of degrees dj , 1 ≤ j ≤ q, in ℙn (ℂ) located in general position. Then, for all ϵ > 0 and 0 < r < R0 , we have q

(q − (n + 1 + ϵ))Tf (r) ≤ ∑ dj−1 Nf(l−1) (r, Dj ) + S(r), j=1

2

where l ≤ 2n +4n en d 2n (nI(ϵ−1 ))n , d = l.c.m.{d1 , . . . , dq }, and S(r) is evaluated as follows: (1) In the case R0 < ∞, S(r) ≤ K(log+

1 + log+ Tf (r)) R0 − r

for every r ∈ [0, R0 ) excluding a set E such that ∫E (2) In the case R0 = ∞,

1 dt R0 −t

< ∞.

S(r) ≤ K(log+ Tf (r) + log r) for every r ∈ [0, ∞) excluding a set E ′ such that ∫E′ dt < ∞.

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Index Ahlfors–Schwarz Lemma 28 algebraic curve 38 Algebraic Second Main Theorem 38 associated maps 121 Bernstein’s theorem 59 Brill–Segre formula 43 Brody hyperbolic 133 Cartan’s characteristic function 90 Cartan’s Second Main Theorem 95 Cartan’s Second Main Theorem for holomorphic curves 89 Catenoid 47 characteristic function 77 Chern–Osserman 135 counting function 76, 91 defect relation 99 degenerated second main theorem 119 divergent curve 58 dual form of the Ahlfors–Schwarz–Pick lemma 32 Enneper surface 48 Euler’s Formula 5 first fundamental form 2 First Main Theorem 77 Fubini–Study form 122 Fujimoto’s five point theorem 61 Gauss curvature estimate 145 Gauss map 3 Gauss’ theorem egregium 8 Gauss–Bonnet for pseudo-metric 35 Gauss–Bonnet theorem 14 Gaussian curvature 5 General Second Main Theorem 98 generalized Gauss map 139 geodesic curvature 3 harmonic function 24 Heinz’s theorem 60 helicoid 47 holomorphic curves 90 holomorphic function 19 Hurwitz’s theorem 24 https://doi.org/10.1515/9783110989557-008

hypersurface 99 in general position 93, 99 isothermal parameters 136 isothermal parametrization 50 Jensen’s formula 76 k-th associated map 38, 121 Lemma on the Logarithmic Derivative 84 linearly non-degenerate 39, 121 linearly nondegenerate 168, 169 Liouville’s theorem 23 little Picard theorem 32 logarithmic derivative in several complex variables 164 mean curvature 5 meromorphic map 167 metric form 35 metric with negative curvature 127 minimal surface 47, 136 minimal surface with finite total curvature 70 n-subgeneral position 113–115 negative curvature method 25, 132 Nevanlinna functions 76 Nevanlinna theory of meromorphic functions 74 Nevanlinna Theory on the Open Ball 160 Nevanlinna’s defect 175 Nirenberg’s Conjecture 60 Nochka constant 115 Nochka diagram 113 Nochka polygon 113 Nochka weight 115 Nochka’s theorem 115 non-integrated defect 174 normal curvature 3 Osserman’s theorem 61 Picard’s theorem 88 Plücker formula 42 Poincaré metric 28 Poisson–Jensen Formula 74 product to sum estimate 127, 128

194 � Index

projective distance 122 proximity function 77, 91 pseudo-metric 27

SMT for algebraic curves with truncation 45 stereographic projection 57 surface immersed in ℝm 136

reduced representation 38, 90, 121, 167, 169 result of S. T. Yau 64 Ricci form 35 Riemann surface 35 Riemann–Hurwitz 36 Ru’s theorem 99, 109

the degenerated case 112 truncated counting function 91

Scherk surface 48 second fundamental form 4 Second Main Theorem 85, 173

value distribution of Gauss map 140 Weierstrass–Enneper representation 55, 56 Weil function 90 Xavier’s theorem 64

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