Seminar On Minimal Submanifolds. (AM-103), Volume 103 9781400881437

Table of contents :
CONTENTS
INTRODUCTION
SURVEY LECTURES ON MINIMAL SUBMANIFOLDS
ON THE EXISTENCE OF SHORT CLOSED GEODESICS AND THEIR STABILITY PROPERTIES
EXISTENCE OF PERIODIC MOTIONS OF CONSERVATIVE SYSTEMS
ARE HARMONICALLY IMMERSED SURFACES AT ALL LIKE MINIMALLY IMMERSED SURFACES?
ESTIMATES FOR STABLE MINIMAL SURFACES IN THREE DIMENSIONAL MANIFOLDS
REGULARITY OF SIMPLY CONNECTED SURFACES WITH QUASICONFORMAL GAUSS MAP
CLOSED MINIMAL SURFACES IN HYPERBOLIC 3-MANIFOLDS
MINIMAL SPHERES AND OTHER CONFORMAL VARIATIONAL PROBLEMS
MINIMAL HYPERSURFACES OF SPHERES WITH CONSTANT SCALAR CURVATURE
REGULAR MINIMAL HYPERSURFACES EXIST ON MANIFOLDS IN DIMENSIONS UP TO SIX
AFFINE MINIMAL SURFACES
THE MINIMAL VARIETIES ASSOCIATED TO A CLOSED FORM
NECESSARY CONDITIONS FOR SUBMANIFOLDS AND CURRENTS WITH PRESCRIBED MEAN CURVATURE VECTOR
APPROXIMATION OF RECTIFIABLE CURRENTS BY LIPSCHITZ Q VALUED FUNCTIONS
SIMPLE CLOSED GEODESICS ON OVALOIDS AND THE CALCULUS OF VARIATIONS
ON THE GEHRING LINK PROBLEM
CONSTRUCTING CRYSTALLINE MINIMAL SURFACES
REGULARITY OF AREA-MINIMIZING HYPERSURFACES AT BOUNDARIES WITH MULTIPLICITY
NEW METHODS IN THE STUDY OF FREE BOUNDARY PROBLEMS
SOME PROPERTIES OF CAPILLARY FREE SURFACES
BERNSTEIN CONJECTURE IN HYPERBOLIC GEOMETRY

Citation preview

Annals of Mathematics Studies Number 103

SEMINAR ON MINIMAL SUBMANIFOLDS

ED ITED B Y

ENRICO BOMBIERI

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1983

Copyright © 1983 by Princeton University Press A L L RIGHTS RESERVED

The Annals o f Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: Phillip A. Griffiths, Stefan Hildebrandt, and Louis Nirenberg

ISBN 0-691-08324-X (cloth) ISBN 0-691-08319-3 (paper)

Printed in the United States o f America by Princeton University Press, 41 William Street Princeton, New Jersey

☆

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Library o f Congress Cataloging in Publication data will be found on the last printed page o f this book

CONTENTS

IN T R O D U C T IO N

v ii

S U R V E Y L E C T U R E S O N M IN IM A L S U B M A N IF O L D S L . Simon

3

O N T H E E X I S T E N C E O F S H O R T C L O S E D G E O D E S IC S A N D T H E IR S T A B I L I T Y P R O P E R T I E S W. B allm an , G. T h o rb e rg s s o n , and W. Z i l le r

53

E X I S T E N C E O F P E R IO D IC M O T IO N S O F C O N S E R V A T I V E SYSTEMS H. G lu ck and W. Z i l le r

65

A R E H A R M O N I C A L L Y IM M E R S E D S U R F A C E S A T A L L L I K E M IN I M A L L Y IM M E R S E D S U R F A C E S ? T . K lotz Milnor

99

E S T IM A T E S F O R S T A B L E M IN IM A L S U R F A C E S IN T H R E E D IM E N S IO N A L M A N IF O L D S R . Schoen

111

R E G U L A R I T Y O F S IM P L Y C O N N E C T E D S U R F A C E S W ITH Q U A S IC O N F O R M A L G A U SS M A P R. Schoen and L . Simon

127

C L O S E D M IN IM A L S U R F A C E S IN H Y P E R B O L I C 3 -M A N IF O L D S K. U h le n b e c k

147

M IN IM A L S P H E R E S A N D O T H E R C O N F O R M A L V A R I A T I O N A L PROBLEM S K. U h le n b e c k

169

M IN IM A L H Y P E R S U R F A C E S O F S P H E R E S W ITH C O N S T A N T SCALAR CURVATURE C . -K . P e n g and C . -L . T ern g

177

R E G U L A R M IN IM A L H Y P E R S U R F A C E S E X IS T O N M A N IF O L D S IN D IM E N S IO N S U P T O SIX J. P itts

199

A F F I N E M IN IM A L S U R F A C E S C . -L . T e rn g

207

T H E M IN IM A L V A R I E T I E S A S S O C IA T E D T O A C L O S E D F O R M F . R e e s e H arvey and H. B la in e L a w s o n , Jr.

217

v

vi

CONTENTS

N E C E S S A R Y C O N D IT IO N S F O R S U B M A N IF O L D S A N D C U R R E N T S W IT H P R E S C R IB E D M E A N C U R V A T U R E V E C T O R R. G u lliv e r

225

A P P R O X IM A T IO N OF R E C T IF IA B L E C U R R E N T S B Y L I P S C H I T Z Q V A L U E D F U N C T IO N S F . J. A lm gren, Jr.

243

S I M P L E C L O S E D G E O D E S IC S O N O V A L O ID S A N D T H E C A L C U L U S O F V A R IA T IO N S M. S. B e rg e r

261

O N T H E G E H R IN G L I N K P R O B L E M E. B om bieri and L . Simon

271

C O N S T R U C T I N G C R Y S T A L L I N E M IN IM A L S U R F A C E S J. T a y lo r

275

R E G U L A R I T Y O F A R E A -M IN IM IZ IN G H Y P E R S U R F A C E S A T B O U N D A R I E S W ITH M U L T I P L I C I T Y B . White

293

N E W M E T H O D S IN T H E S T U D Y O F F R E E B O U N D A R Y PROBLEM S D . K inderlehrer

303

SOM E P R O P E R T I E S O F C A P I L L A R Y F R E E S U R F A C E S R. F in n

323

B E R N S T E I N C O N J E C T U R E IN H Y P E R B O L I C G E O M E T R Y S. P . W ang and S .W . W ei

339

IN T R O D U C T IO N

T h e present volum e c o lle c t s the p apers w hich w ere presented in the academ ic year 1979-1980 at the Institute for A d v a n ced Study, in the areas of c lo s e d g e o d e s ic s and minimal s u r fa c e s , a s part of the a c t iv itie s of a s p e c ia l y ear in d iffe re n tia l geom etry and d iffe re n tia l eq u atio n s. Starting w ith a su rvey lectu re, they have been arranged acc o rd in g to dim ension and approach, from c l a s s i c a l to that o f geom etric m easure theory. We w is h to extend our s in c e re thanks to a l l contributors, p articu larly for their c o lla b o ra tio n in se n d in g their texts and their re v is io n s a s w e ll as for their p atien ce in w a itin g for th ese notes to ap pear.

Our thanks a ls o to

the N a tio n a l S c ien c e Foun datio n for supporting this s p e c ia l year at the Institute for A d v a n c e d Study. E N R IC O B O M B IE R I

Seminar On Minimal Submanifolds

S U R V E Y L E C T U R E S O N M IN IM A L S U B M A N IF O L D S L e o n Sim on*

Our aim here is to g iv e a gen e ra l (but n e c e s s a rily b r i e f ) introduction to the theory of minimal su b m a n ifo ld s, in clu d in g as many exam p les a s p o s s ib le , and in clu d in g some d is c u s s io n of the c l a s s i c a l problem s (B e r n s t e in ’s , P la t e a u ’s ) w hich h ave provided much of the m otivation for the developm ent of the theory. Our firs t ta sk is to d is c u s s a notion of minimal “ v a r ie t y ” in a s u f f i­ cien tly gen e ra l s e n s e to in clu de the vario u s c l a s s e s of o b je c ts (e .g . a lg e b r a ic v a r ie t ie s in

C n , co n es over smooth su bm an ifo lds of

“ so a p film li k e ” minimal s u r fa c e s in

Sn ,

R 3 , branched minimal im m ersions,

le a s t area in tegral current re p resen ta tiv e s of hom ology c l a s s e s ) w hich a r is e naturally.

T h is w i l l b e done in §2, after som e c l a s s i c a l introductory

d is c u s s io n of firs t and seco n d variatio n in §1.

In §3 w e present som e of

the p rin cip al c l a s s e s of exam p les of minimal v a rie tie s .

§4

in clu des a

d is c u s s io n of som e of the s p e c ia l properties of minimal su bm an ifo lds of Rn .

In §5 w e g iv e a b rie f survey of the known interior regularity theory,

and in

§6

w e d is c u s s the c l a s s i c a l B ern stein and P la t e a u problem s and

the p resent state of kn ow led ge concern in g them.

F in a lly , in §7, w e d is c u s s

som e s e le c te d a p p lic a tio n s of seco n d variatio n of minimal su bm an ifo ld s in geom etry and to p ology.

^R esearch w as p artially supported by an N .S .F . grant at the Institute for A d v an ced Study, Princeton. © 1983 by Princeton U n iversity P r e s s S e m in a r on M in im a l S u b m a n ifo ld s 0-691-08324-X/83/003-50 $3.00/0 (clo th ) 0-691-08319-3/83/003-50 $3.00/0 (p ap e rback ) F o r copying information, s e e copyright page.

3

4

LEON SIMON

We w ould here like to recommend the survey a rtic le s [ L I ] , [ N l ] , [0 1 ], [B ], w hich cover topics only touched upon (or not mentioned at a l l ) here. F o r other gen eral re ad in g w e stron gly recommend the works [F H 1 ], [G E 1 ], [L 3 ], [L 4 ], [N 3 ], [0 2 ]. In a ll that fo llo w s ,

N

w ill denote an n-dim en sio n al R iem annian

m anifold without boundary U

(n >

2

),

and

k is an integer with

w ill a lw a y s denote an open s u b s e t of

N.

to co n sid er a “ smooth defo rm atio n ” (i.e . a smooth iso to p y ) of holds everything o u tside a com pact s u b s e t of let

(- 1 ,1 ) x N

be a

C2

( - 1 ,1 ) x N ,

^ : N

s u p p o se

N

X

fix ed .

0 o = i N ,

be defin ed by

K CU

N

w hich

T o be s p e c ific , 0 : ( - 1 , 1 ) x N -> N

t(x ) = 0 ( t , x )

is a diffeom orphism of each

s u p p o se there is a com pact

(0 .1 )

U

be equipped with the product metric, let

map, let

< k < n.

1

We a ls o often have o c c a s io n

for

( t , x )

1=1

with

t-

as in ( 1 .4 ).

It go es without s a y in g (and is e a s ily c h e c k e d ) that a ll th ese d efin itio n s are independent of the particu lar ch o ice of

•••,

We now have from (1 .9 ), (1 .1 0 ), and (1 .1 1 ) that

I (1 .1 2 )

= “

d iv MX

X

a=k+l

= - , and hence

J d iv „X

d „ -

f

d l , „ X T d? - I

M

< X , H > d fi .

M

T o go further, w e assum e that

M

is a c tu a lly com pact with smooth boundary

(9M (p o s s ib ly em pty); then the c l a s s ic a l d iv e rg e n c e theorem te lls us (s in c e

2

i.J=l

^

< R (r i,X i )X i , r i > ) d , i .

1=1

In ch eck in g this w e first show that that

= J i

2

a s in (1 .9 ), (1 .1 0 ) a b o v e ).

I (X ) = I ( X ^ )

and then u se the fact

< r - , V v a X X , v a > = - < B ( r - , t -), X > a = k + iJ i '

( va,

B

F o r further d e ta ils concerning this com putation,

SURVEY LECTURES ON MINIMAL SUBMANIFOLDS

s e e for exam ple [S L 1 ]. X

, w here

=

In the cod im ension 1 oriented c a s e , w e can write

is a smooth unit normal and

v

and the e x p re s s io n for

I(X )

I (X ) =

{!V £ | 2 - C

2

k (| B | 2 + ] £

M

w here

V£

k 2

< R (r -,v )v ,r ->

1=1

1

R ic c i curvature of

is a s c a la r function,

< R ( r i , l, V , r i >) } d ^ ,

i= l

d enotes the gradient (tak en in

N o tic e that

£

becom es

c (1 .1 9 )

11

is ju st

M ) of the s c a la r function

Ric(i/, v ) , w here

R ic

£.

is the

1

N .

We should fin a lly mention the m eaning of the terms “ station ary in and “ s ta b le in

U ” in c a s e

this w e can s u p p o se that

M

is immersed rather than em bedded.

(not n e c e s s a rily an im m ersion), and let J 0 / 0 (x ) = ||Aj^(d ^r)|| , where

TXM 0 ->

induced by

if/ .

of co u rse d efin ed by (1 .2 0 )

A( 0 for every su ch variatio n .

Then

K C U Q,

x e MQ ~

is d efin ed on

if/

0

= if/ and su ch that, for some fix ed com pact

U Q if

if/.

J( N

T h e area a s s o c ia t e d with su ch a

M

B y a variation of

T o do

M Q is any com pact k-dim en sio n al Riem annian

m anifold (w ith or without boundary) and let

T h at is ,

U ”

n Xt = i f / f a )| £ dt ^ t v ‘t = 0

T N ) then one can

12

LEON SIMON

d eriv e e x p re s s io n s for firs t and seco n d v ariatio n s (i.e . for d2 — d t2

and

^

^

) w hich are the sam e a s the right s id e s of ( 1 . 5 ), ( 1 .6 ), t= 0

but in w hich the vario u s quan tities must now be appropriately interpreted. (N e a r points hood

W

of

x Q e M Q , w here x Q into

N

J 0 A ) (x o) ^ 0 ,

if/

embeds a sm all n eigh b o r­

and one com putes the form ulae for firs t and

seco n d v ariatio n by com puting the t-d e riv a tiv e s of the relevan t J aco b ian a s befo re.

On the other hand, the points w here

contribute nothing to the fo rm u la e.) sio n , it fo llo w s that if and only if

if/

near points w here

of co u rse

Of co u rse from the previou s d is c u s ­

U Q H dMQ = 0 ,

lo c a lly em beds

J 0 A )(x o) = 0

then

if/

is stationary in

U Q if

M Q a s a ze ro mean curvature subm an ifold

J (if/) ^ 0 .

§2 . k-varifolds, k-currents An exam ination of the d is c u s s io n in §1 w ill show that the form ulae (1 .5 ), (1 .6 ) (and their d e riv a tio n s) remain v alid even in the p resen ce of s erio u s s in g u la ritie s in

M.

T o make this statement p re c is e , w e first

need to introduce some term inology. We henceforth let

M b e a countably k-r e ctifia b le B o r e l s e t in

N .

oo T h at is , C

1

M

is B o re l,

subm an ifolds of

M C U M- , where j= l J N

( Mj

NL are (o p e n ) k-dim en sion al J

not n e c e s s a rily com plete nor p a irw is e

d is jo in t). ^ We a ls o a llo w the introduction of a multiplicity function of s p e c ific a lly , w e let w here

0

p be the m easure on

is a non-n egative lo c a lly

multiplicity function) on rather than ju st on

M,

of co u rse often have

N.

N

d efin ed by

M;

dp = ^ d K ^ ,

K ^-sum m able function (c a lle d the

(We take

6 to be d efined on a ll on

N ,

for re ason s of purely te ch n ical co n ven ien ce; w e

6 = 0 on

N ~

M .)

^ M o d u lo sets of K k -m easure zero, this is equivalent to the requirement that M is contained in a countable union of im ages, under L ip sch itz maps, of compact subsets of R k .

(By [F H 1 , 3 .1 .1 6 ].)

SURVEY LECTURES ON MINIMAL SUBMANIFOLDS

T h e pair

(M, p )

w ill then be c a lle d a k -varifo ld in

N .

13

(T h is co rre ­

spo n d s e x a c tly to the d efin itio n of k -dim en sio n al re c tifia b le v arifo ld as d efin ed by A lla r d [A W 1 ]; one s e e s this by virtue of [AW 1, 3 .5 (1 ), 2 . 8 (5 )]. / * ^ is c a lle d an integral k -varifo ld if the m ultiplicity function

QA, p)

is

0

in tege r-v a lu ed . T he support, denoted (2 .1 )

spt(M , p ) ,

spt(M , p ) = N -

w here the union is over a l l open spt(M , f ) C M

(c lo s u re of

true.

spt(M , p )

In fac t

d efin ed by

/x L M .

p L M (A ) = p Q A f \ A ) . )

1

su ch that

subm an ifold of

denoted

s in g (M ,/ z ),

(2 .2 )

(/xLM

p (M )

is the (o u te r) m easure on N

is c a lle d the mass of

(M , p) .

is d efin ed to be the s et of

x

in

is a properly em bedded k-dim en sion al

for som e open

W

co ntaining

x.

The singular set,

is d efin ed by

sin g (M , p ) = s p t (M , p )

G iv en

N o tic e that

N ), but eq u a lity may be far from

reg(M ,^z),

spt(M , p ) fl W N

^ ( M f lW ) = 0 .

is ju st the support (in the u su a l s e n s e of m easures

T he regular set, denoted spt(M , p)

is d efin ed by

U W

W su ch that

M taken in

[F H 1 , Ch. 2 ]) of the m easure

C

of the k -v arifo ld (M, p )

a k -v arifo ld

(M, p )

in

N , and a smooth map

other Riem annian m an ifo ld ) such that the image varifold i/a#(M, p )

(2 .3 )

reg(M , p ) .

i/r|spt(M, p )

if/ : N

N

( N

any

is p r o p e r / ^ w e d efin e

to be the k -varifo ld in

N

given by

(z).

zcif/ 1(y)riM F or an alternative description of A lla r d ’s work, s ee [S L 5 ]. ^T hat is,

is compact whenever

K C N

is compact.

if/

if/^p 6 (y) =

is

LEON SIMON

14

N o tic e that if

is a diffeom orphism then

if/

m ultiplicity function

^ #(M, p )

has the

6 = 6°ifz~^ .

We then have (by virtue of the area formula for k -re c tifia b le s e ts — [F H 1 , § 3 .2 ]), that (w ith

U

open such that

l0 t# (M n u ,/ z )|

=

ft(M D U ) < °o )

f

] ( t f x )d p

MflU w h en ever

| TXN

v a lidity

if/ one c h o o s e s to u s e ).

fd K k ,

a diffeom orphism

( 2 -4 ) is independent of

O f co u rse

here

d x ifj denotes

if/ .

U s in g stan dard, m easu re-theoretic fa c ts about d e n s itie s ([F H 1 ,

2 .1 0 .1 9 (4 )] and the fact that

alm ost a l l

f e A , in c a s e

meas lim ---------------- p - ------ = 1 P±° m e a s (B ^ ))

AC

for L e b e s g u e

is L e b e s g u e m easu rab le), together

w ith the d efin itio n of coun tably k -re c tifia b le s et given a b o v e, one e a s ily ch e ck s that such an approxim ate tangent s p a c e e x is t s

p - a .e . in

We thus h av e , by e x ac tly the com putations of §1, that

M.

SURVEY LECTURES ON MINIMAL SUBMANIFOLDS

15

^ l * t# ) d ^ .

to be stationary in

U

( re s p e c t iv e ly

U ) if the integral on the right of ( 2 .5 ) v a n is h e s (re s p e c t iv e l y if

the right s id e of (2 . 5 ) v a n i s h e s and the right of (2.6 ) > 0 ) for a l l smooth vector fie ld s

X

w hich have compact support in

U.

N o t ice a l s o that a g a in the first term on the right of ( 2 .6 ) ca n a l w a y s be d ele ted in c a s e

(M , n )

is stationary in

U.

Fo r later ap p lic a tio n s it is conven ient here to introduce a s lig h t gen eraliz atio n of the notion of “ s ta t io n a r y ’ ’ . has g en e ra liz ed m ean-curvature v ecto r

I

d i v MXd/x = -

M

whenever

X

Evid ently

(M, /i)

M

P

in U

if

H c L^/i, U )

and if

M

is statio nar y in U,

U

U.

if and only if it has z e ro gen eraliz ed

and (by the c l a s s i c a l formula (1.13 )), in

is a compact smooth manifold with

dMHU = 0

fi = K k L M ), the g e n e ra liz e d mean curvature vector of c o in c id e s with the c l a s s i c a l mean curvature vector of on

(M,/x)

< X , H > d ii

is a smooth vect or field with compact support in

mean curvature vector in case

H

N am ely, w e s a y that

(and in c a s e (M ,K k L M )

M a s a submanifold

N . Our main reason for introducing this notion is the fo llo w in g remark.

LEON SIMON

16

REMARK 2.1.

If

N

is (without lo s s of generality, by [ N J ] ) as sum ed to

be is omet rically embedded

in RP ,

consid ered as a k-varifold

in N ,

k-varifold in

RP ,

at each point

x

(M, fi)

of

has

U C RP

is open,

is stationary in

and

U fl N ,

if

then,

gen eraliz ed meancurvature vector

(M, f i ) , as a

H which

M fl U s a t i s f i e s |H| < k|BN | ,

(2.7 )

where

|BN | denotes the length of the second fundamental form of

a subm anifold of

X

on N

vecto r fie ld on

N

(as

RP ).

T o s e e th is, w e note that fie ld

if

f

M

d iv MX dfj. = 0 for any smooth vecto r

with com pact support in

U H N . Thus if

RP with com pact support in

(2 .8 )

I

U,

X

is a smooth

then

d i v u ( X - X 1 ) d fi = 0 ,

M w here

X^

.

P

X

=

denotes the component of

2 i s a , where a -n + l

normal s e t of vecto rs in

RP

X

normal to

N . Then (s in c e

w e have

a -

Thus, lo c a lly ,

i/P is a sm oothly varyin g ortho-

normal to

i= l

N.

n+1

k

P

i=l

a=n+l k

-< x . 2 V i ^ i ^ 1=1

= 0 ),

SURVEY LECTURES ON MINIMAL SUBMANIFOLDS

w here

BN

is the seco n d fundam ental form on

N

17

a s a subm an ifold of

RP.

T h u s (b y (2 .8 ))

J d i v MX

= - J d fi ,

k and hence

B lsJ(V -,r -)

2

i= l QA,fi)

1

is the ge n e ra liz e d mean curvature vector of

1

(a s a k -v arifo ld in

RP ) on

U.

(2 .7 ) evid en tly fo llo w s .

We w ill a ls o have o c c a s io n s u b seq u e n tly to co n sid er k-currents. k-current in

N

is sim ply an oriented integral k -varifo ld in

an orientation a

•••

£. at

a

H ere, by an orientation ^-alm o st a ll points

x (M

£ w e mean that

is

f(x ) =

(w here

orthonormal b a s is for the approxim ate tangent s p a c e £

That is , a

(M, (j l ) , with in tege r-va lu ed m ultiplicity function, together with

k-v arifo ld

± ^

N.

A

is any TXM ), and a ls o that

^ -m easu rab le w hen co n sid ered lo c a lly (v ia coordinate charts of

a s a m apping from

W fl M

( W

a coordinate neighborhood of

N )

N ) into

A k (R n) . The notion of k-current c o in c id e s exa c tly with the notion of k -d im en sio n al lo c a lly re c tifia b le current in the s e n s e of F e d e re r and F le m in g [ F F ] , and F e d e re r [ F H l ] .

T h is becom es cle a re r (it is form ally

proved from [ F H l , 4 .1 .2 8 ]) when w e note that, a s s o c ia t e d w ith each k-current

(

M

smooth k-forms

i

n

N,

w e have a lin ear fu n ction al

co with com pact support in

(2 .9 )

T(fi>) =

I

N

T

d efin ed on the

by

< (* ,£ > d m

M

here

( x ) = < t u ( x ) , f ( x ) >

denotes the d u al p airin g betw een

k-co vectors and k-vectors on

TXN .

thus id en tify in g the k-current

(M,/z, £ )

tion al on k-form s.

We w ill often w rite

T = ( M , / * ,£ ),

w ith the a s s o c ia t e d linear fu n c ­

B y an a lo gy w ith S to k e s ’ Theorem , w e can d efin e

LEON SIMON

18

another fu n ctio n al (c a lle d the boundary of

(2.10)

T ) by

(< 3 T )(w ) = T ( d o )

for a l l smooth (k -l)-fo r m s

co w ith compact support in

N.

N o tic e that

(9 T , s o d efin ed , may or may not represent a (k -l)-c u r r e n t on

N.

In fact

there is a very u s e fu l theorem due to F e d e re r and F lem in g ( [ F H l , 4 .2 .1 6 ]) w hich a s s e rt s that if is fin ite and if

T = (M, p, f )

is a k-current and if the mass of

dT

spt