Harmonic Maps and Minimal Immersions with Symmetries (AM-130), Volume 130: Methods of Ordinary Differential Equations Applied to Elliptic Variational Problems. (AM-130) 9781400882502

Table of contents :
Table of Contents
INTRODUCTION
PART I Basic Variational and Geometrical Properties
Chapter I Harmonic maps and minimal immersions
Introduction
Basic properties of harmonic maps
Minimal immersions
Notes and comments
Chapter II Immersions of parallel mean curvature
Introduction
Parallel mean curvature
Alexandrov’s theorem
Notes and comments
Chapter III Surfaces of parallel mean curvature
Introduction
Theorems of Chem and Ruh–Vilms
Theorems of Almgren–Calabi and Hopf
On the Sinh–Gordon equation
Wente’s theorem
Notes and comments
Chapter IV Reduction techniques
Introduction
Riemannian submersions
Harmonic moiphisms and maps into a circle
Isoparametric maps
Reduction techniques
Notes and comments
PART II G–Invariant Minimal and Constant Mean Curvature Immersions
Chapter V First examples of reductions
Introduction
G–equivariant harmonic maps
Rotation hypersurfaces in spheres
Constant mean curvature rotation hypersurfaces in R^n
Notes and comments
Chapter VI Minimal embeddings of hyperspheres in S^4
Introduction
Derivation of the equation and main theorem
Existence of solutions starting at the boundary
Analysis of the O.D.E. and proof of the main theorem
Notes and comments
Chapter VII Constant mean curvature immersions of hyperspheres into R^n
Introduction
Statement of the main theorem
Analytical lemmas
Proof of the main theorem
Notes and comments
PART III Harmonic Maps Between Spheres
Chapter VIII Polynomial maps
Introduction
Eigenmaps S^m → S^n
Orthogonal multiplications and related constructions
Polynomial maps between spheres
Notes and comments
Chapter IX Existence of harmonic joins
Introduction
The reduction equation
Properties of the reduced energy functional J
Analysis of the O.D.E.
The damping conditions
Examples of harmonic maps
Notes and comments
Chapter X The harmonic Hopf construction
Introduction
The existence theorem
Examples of harmonic Hopf constructions
π(S^2) and harmonic morphisms
Notes and comments
Appendix 1 Second variations
Appendix 2 Riemannian immersions S^m → S^n
Appendix 3 Minimal graphs and pendent drops
Appendix 4 Further aspects of pendulum type equations
References
Index

Citation preview

Annals of Mathematics Studies

Number 130

Harmonic Maps and Minimal Immersions with Symmetries M

ethods of

A

p p l ie d to

O

r d in a r y

D

E l l ip t ic V

if f e r e n t ia l a r ia t io n a l

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q u a t io n s

P roblem s

by

James Eells and Andrea Ratto

P R IN C E T O N U N IV E R S IT Y P R E S S P R IN C E T O N , N E W JE R S E Y 1993

Copyright © 1993 by Princeton University Press ALL RIGHTS RESERVED Printed in the United States o f America The Annals o f Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability o f the Committee on Production G uidelines for Book Longevity o f the Council on Library Resources

L ibrary of Congress Catalog-in-Publication Data E ells, James, 1926Harmonic maps and minimal immersions with symmetries : methods o f ordinary differential equations applied to elliptic variational problems / by James Eells and Andrea Ratto. p.

cm. — (Annals o f mathematics studies ; no. 130) Includes bibliographical references and index.

ISBN 0-691-03321-8 — ISBN 0-691-10249-X (pbk.) 1. Harmonic maps. 2. Immersions (M athematics). 3. Differential equations, E lliptic— Numerical solutions. I. Ratto, Andrea, 1961-. Q A 614.73.E35 514’.7— dc20

1993 92-38760

Harmonic Maps and Minimal Immersions with Symmetries

HARMONIC MAPS AND MINIMAL IMMERSIONS WITH SYMMETRIES

Methods of ordinary differential equations applied to elliptic variational problems

James Eells and Andrea Ratto

INTRODUCTION In this monograph we study harmonic maps, minimal and parallel mean curvature immersions in various symmetric contexts. The maps under consideration are solu­ tions to certain elliptic variational problems; and in the presence of suitable symme­ try, those often admit reductions to lower dimensional problems whose qualitative study is more manageable. (1) Our reduction theory (Chapter IV) is based on the differential geometry of fibre bundles - just as our variational theory is expressed in terms of the calculus in Riemannian vector bundles. For a start: A variation (pt)t£R of a map po = p : M -+ N between two Rieman­ nian manifolds determines a section t?= — I dt u=o of the induced vector bundle p ~ lT ( N ) —►M (defined in Chapter I (1.1)). We can interpret the differential dp : T ( M ) T ( N ) as a section of the tensor product T*(M) 0 p ~ lT ( N ) -* M y where T*(M) denotes the cotangent bundle of M. All of these vector bundles have Riemannian structures (i.e., a metric a on the fibres, together with a linear connection V for which Va = 0) induced from the metrics of M and N. The second fundamental form of p is the section of G)2T * ( M ) 0 p ~ lT ( N ) given

by

(2)

(Vd