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

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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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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