Plateau's Problem and the Calculus of Variations

  • 96 43 2
  • Like this paper and download? You can publish your own PDF file online for free in a few minutes! Sign Up

Plateau's Problem and the Calculus of Variations

by Michael Struwe Mathematical Notes 35 PRINCETON UNIVERSITY PRESS PRINCETON) NEW JERSEY 1988 Copyright © 1989

423 19 9MB

Pages 158 Page size 415.2 x 653.04 pts Year 2011

Report DMCA / Copyright

DOWNLOAD FILE

Recommend Papers

File loading please wait...
Citation preview

PLATEAU'S PROBLEM AND THE CALCULUS OF VARIATIONS

by

Michael Struwe

Mathematical Notes 35

PRINCETON UNIVERSITY PRESS

PRINCETON) NEW JERSEY

1988

Copyright © 1989 by Princeton University Press All Rights Reserved

Printed in the United States of America by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

The Princeton Mathematical Notes are edited by William Browder, Robert Langlands, John Milnor, and Elias M. Stein

Library of Congress Cataloging in Publication Data

Struwe, Michael, 1955Plateau's problem and the calculus of variations. (Mathematical notes; 35) Bibliography: p. 1. Surfaces, Minimal. 2. Plateau's problem.

3. Global analysis (Mathematics) 4. Calculus of variations. I. Title. II. Series: Mathematical notes (Princeton University Press) ; 35. QA644.S77

1988

516.3'62

ISBN 0-691-08510-2 (pbk.)

88-17963

To Anne

Contents A. The "classical" Plateau problem for disctype minimal surfaces.

I.

Existence of a solution 1. The parametric problem 2. A variational principle 3. The direct methods in the calculus of variations 4. The Courant-Lebesgue Lemma and its consequences 5. Regularity Appendix

II.

5 7 12

16 22 29

Unstable minimal surfaces 1. 2. 3. 4. 5. 6.

Ljusternik-Schnirelman theory on convex sets in Banach spaces The mountain-pass lemma for minimal surfaces Morse theory on convex sets Morse inequalities for minimal surfaces Regularity Historical remarks

33 41

52 60 66 78

B. Surfaces of prescribed constant mean curvature.

III.

The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in m? 1. 2. 3. 4. 5.

IV.

The variational problem The volume functional "Small" solutions Heinz' non-existence result Regularity

Unstable H -

91 94 100 104 105

surfaces

1. H - extensions 2. Ljusternik-Schnirelman and Morse theory for "small" H - surfaces 3. Large solutions to the Dirichlet Problem 4. Large solutions to the Plateau problem: "Rellich's conjecture"

111 116 121 127

References

141

ix

Preface

Minimal surfaces and more generally surfaces of constant mean curvature - commonly known as soap films and soap bubbles - are among the oldest objects of mathematical analysis. The fascination that likewise attracts the mathematician and the child to these forms may lie in the apparent perfection and sheer beauty of these shapes. Or it may rest in the contrast between the utmost simplicity and endless variability of these remarkably stable and yet precariously fragile forms. The mathematician moreover may use soap films as a simple and beautiful model for his abstract ideas. In fact, long before the famous experiments of Plateau in the middle of the 19th century that initiated a first "golden age" in the mathematical study of minimal surfaces and through which Plateau's name became inseparably linked with these objects, Lagrange investigated surfaces of least area bounded by a given space curve as an illustration of the principle now known as "Euler-Lagrange equations". However, for a long time since Lagrange's derivation of the (non-parametric) minimal surface equation and Plateau's soap film experiments ma~hematicians had to acknowledge that their methods were completely inadequate for dealing with the Plateau problem in its generality. In spite of deep insights into the problem gained by applying the theory of analytic functions the solution to the classical Plateau problem evaded 19th century mathematicians- among them Riemann, Weierstrafi, H.A. Schwarz. To meet the challenge ideas from complex analysis and the calculus of variations had to merge in the celebrated papers by J.Douglas and T. Rad6 in 1930/31. But this was only the beginning of a new era of minimal surface theory in the course of which many significant contributions were made. Among other discoveries it was noted that the (parametric) Plateau problem may possess unstable solutions - which of course are not seen in the physical model - and in particular that the solutions to the Plateau problem in general are not unique. The question whether for "reasonable" boundary data the Plateau problem will always have only a finite number of solutions still puzzles mathematicians today. The most significant contributions are Tomi's result on the finiteness of the number of surfaces of absolutely minimal area spanning an analytic curve in IR3 and the generic finiteness result of Bohme and Tromba. The existence of "unphysical" solutions in the parametric problem in the 60's led to a new approach to the Plateau problem by what is now known as "geometric measure theory". In the course of these developments the notions of surface, area, tangent space, etc. came to be reconsidered and the notion of "varifold" evolved which parallels the notion of "weak solutions" in partial differential equations, cpo Nitsche [I, § 2].

x

Plateau Problem - Preface

But also the theory of the parametric Plateau problem was further pursued. Both to bring out the geometric content of the parametric solutions obtained (branch points, self-intersections) and - independently of the physical model- to explore the richness of a fascinating variational problem in its own right. In this monograph we will focus our attention on the interplay between the parametric Plateau problem and developments in the calculus of variations, in particular global analysis. For reasons of space we will at most casually touch upon the more geometric aspects of the problem. As far as classical results about the geometry of minimal surfaces or the geometric measure theory approach to minimal surfaces are concerned the reader will find ample material and references in J .C.C. Nitsche's encyclopedic book [1] or in the lecture notes by L. Simon [1]. Our main emphasis will be on the power of the variational method.

Notations • denotes duality; occasionally we also denote a certain normalization with an asterisque. These notes are divided into two parts with together four chapters, each divided into sections. Sections are numbered consecutively within each chapter. In crossreferences to other chapters the number of the section is preceded by the number of the chapter which otherwise will be omitted. These notes are based on lectures given at Louvain - La - Neuve and Bochum in 1985-86. I am in particular indebted to Reinhold Bohme, Stefan Hildebrandt, Jean Mawhin, Anthony Tromba, Michel Willem and Eduard Zehnder for their continuous interest in the subject which has been a major stimulus for my work. Special thanks I also owe to Herbert Graff for his diligence and enthusiasm at typesetting this manuscript with the AMS-'!EXsystem. Finally, I wish to express my gratitude for the generous support of the SFB 72 at the University of Bonn.

Michael Struwe

Zurich, March 1988

PLATEAU'S PROBLEM AND THE CALCULUS OF VARIATIONS

A. The "classical" Plateau problem for

disc-type minimal surfaces

I. Existence of a solution. 1. The parametric problem. Let r be a Jordan curve in JR.". The "classical" problem of Plateau asks for a disc-type surface X of least area spanning r; Necessarily, such a surface must have mean curvature O. If we introduce isothermal coordinates on X (assuming that such a surface exists) we may parametrize X by a function X(w) (Xl{w), ... , X"{w)) over the disc

=

satisfying the following system of nonlinear differential equations

(1.1) (1.2) (1.3)

b.X = 0 in B,

= =

IXul2 -IXtJ I2 0 Xu' XtJ in B, XI8B : 8B -+ r is an (oriented) parametrization of

Here and in the following Xu Euclidian JR." .

= -luX,

r.

etc., and . denotes the scalar product in

Conversely, a solution to (1.1) - (1.3) will parametrize a surface of vanishing mean curvature (away from branch points where VX(w) 0) spanning the curve r, i.e. a surface satisfying the required boundary conditions and whose surface area is stationary in this class. Thus (1.1) - (1.3) may be considered as the Euler Lagrange equations associated with Plateau's minimization problem.

=

However, (1.1) - (1.3) no longer require X to be absolutely area-minimizing. Correspondingly, in general solutions to (1.1) - (1.3) may have branch points, selfintersections, and be physically unstable - properties that we would not expect to observe in the soap film experiment. Thus as we specify the topological type of the solutions and relax our notion of "minimality" a new mathematical problem with its own characteristics evolves. In the following we simply refer to solutions of (1.1) -(1.3) as minimal surfaces spanning r. In this first chapter we present the classical solution to the parametric problem (1.1) - (1.3). Later we analyze the structure of the set of all solutions to (1.1) -(1.3). The key to this program is a variational principle for (1.1)-(1.3) which is "equivalent" to the least area principle but is not of a physical nature as it takes account of a feature present in the mathematical model but not in the physical solution itself: The parametrization of a solution surface. This variational principle is derived in the next section. Applying the "direct methods in the calculus ofvariations" we then

6

A. The classical Plateau Problem for disc _ type minimal surfaces.

obtain a (least area) solution to the problem of Plateau. At this stage the CourantLebesgue-Lemma will be needed. Finally, some results on the geometric nature of (least area) solutions will be recalled.

=

It will often be convenient to use complex notation and to identify points w (u,v) E B with complex numbers w u+iv rei~ E ([;'. Moreover, we introduce the complex conjugate 10 u - iv and the complex differential operators

=

=

=

Note that 88 =~; hence any solution X to (1.1) - (1.2) gives rise to a holomorphic differential 8X : B C ([;' --+ ([;' n satisfying the conformality relation

8X 2

= f: (8Xi)2 = 0,

cpo Lemma 2.3. Conversely, from any holomorphic curve

i=l

F : B C ([;' -+ ([;' n

satisfying the compatibility conditon

F2

=0

a solution

tIJ

X(w) == Real

J F dw

to (1.1), (1.2) may be constructed.

This relation between minimal surfaces and holomorphic curves is the basis for the classical WeierstraB - Enneper representations of minimal surfaces in !If which constitute one ofthe major tools for constructing and investigating minimal surfaces, cpo Nitsche [1, §§ 155 - 160] .

7

I. Existence of a solution.

Let H 1 ,2 (B i JRn) be the Sobolev space of with square integrable distributional derivatives,

2. A variational principle. L 2 -functions X: B --+ JRn and let

J

IXI 2 dw

B

JIVXI2dw IIXII~ = IIXII~ + IXI~ = J(IXI 2+ IVXI2) IXI~ = IIVXII~ =

B

dw

B

denote the

L 2 -norm,

respectively the seminorm and norm in H 1 ,2(Bi JRn).

A(X)

=

Jv'IXuI2IX" 12 -IXu

.X,,1 2 dw

B

X, cpo Simon [1, p. 46].

denote the area of the "surface" Also introduce the class

c(r)

= {X E H 1 ,2(Bi JRn ) I XI8B E CO(8B,JRn )

of H 1 . 2 -surfaces spanning

is a weakly monotone parametrization of

r}

r.

Note that the area of a surface X does not depend upon the parametric representation of X, i.e.

(2.1)

A(X 0 g)

= A(X)

for all diffeomorphisms 9 of B. Hence by means of the area functional it is impossible to distinguish a particular parametrization of a surface X, and any attempt to approach the Plateau problem by minimizing A over the class C(r) is doomed to fail due to lack of compactness. In 1930/31 Jesse Douglas and Tibor Rad6 however ingenuously proposed a different variational principle where the minimization-method meets success: They (essentially) considered Dirichlet's integral

D(X)

=

1/2

JIVXI2dw B

instead of A. For this functional the group of symmetries is considerably smaller; the relation

(2.2)

D(X 0 g)

= D(X)

A. The cl....sical Plateau Problem for disc - type minimal surfaces.

8

9 of

only holds for conformal diffeomorphisms 9 satisfying the condition

Ig.. 12

(2.3)

Now, A and

Ig.. 12

-

= 0 = g.. ' g.,

B, i.e. for diffeomorphisms

in B.

D are related as follows: For X E H I ,2(Bj IRn)

(2.4)

A(X)

~

D(X)

with equality iff X is conformal, i.e. satisfies (1.2). Conversely, given a surface parametrized by X E H I ,2(BjIRn ) we can assert the following result due to Money [2j Theorem 1.2]: Theorem 2.1: Let X E H I ,2(Bj IRn ), E > O. There exists a diffeomorphism 9 : B - B such that X' X 0 9 satisfies:

=

D(X')

~

(1 + E) A(X')

(1 + E) A(X).

In particular, Theorem 2.1 implies that inf

(2.5)

XEC(r)

A(X)

= XEC(r) inf D(X).

We will not prove Money's E-conformality result. However, with the tools developed in Chapter 4 it will be easy to establish (2.5) for rectifiable r, cpo the appendix. By (2.5), for the purpose of minimizing the area among surfaces in C(r) it is sufficient to minimize Dirichlet's integral in this class. Moreover, we have the following Lemma 2.2: X E C(r) solves the Plateau problem (1.1) - (1.3) iff critical for D on C(r) in the sense that

ii)

X

is

£D(X 0 gelj BdlE=o = 0 for any family of diffeomorphisms gE: B BE depending differentiably on a parameter lEI < EO, and with go = id .

Proof:

Compute

~D(X +E1,1/>2,1/>3), (1/J1,1P2,1/J3), 0 ~ 1,01 < < 1/J2 < 1/J3 < 211", there exists a unique 9 E G such that

1,02


1 . ,. ] = 1 2"3. ( irp.) = e ge1

Lemma 4.2 suggests to normalize admissible functions as follows: Let ~.

=

Pj

=

e 3 ,] 1,2,3 and let Qj, j 1,2,3 be an oriented triple of distinct points on f. Define C*(f) {X E C(f)IX(Pj) Qj, j 1,2,3}.

=

=

=

Then we obtain the following crucial result:

Lemma 4.3 The injection C* (f) -+ CO( oB; JR") bounded subsets of C*(f) are equicontinuous on oB.

is compact, i.e.

D-

For the proof we need the following fundamental lemma due to Courant [1, p. 101

fr.] and Lebesgue [1, p. 388]: Lemma 4.4: For any X E H 1 ,2(B;JR"), any wE B, any exists p E [6, vb] such that if s denotes arc length on

Cp

6 E ]0, 1[ there

= Cp(w) = oBp(w) n B

we have: X. E L2(Cp ) and

JIX;lds ~

8D(X)/plln pl·

cp Proof:

By Fubini's theorem

IX.I E L2(Cp)

for a.e.

p

< 1 and

../6

2D(X) >

J IVXI 2dw (B ../6(1D )\B6(1D »nB essinf 6$.p$.../6

(p J Ix. 12 dS) Cp

~J 6

J Cp

.J../6dpp. 6

Since for all p E [6, vb] ../6

J

dpP -- 1/2 lIn 61 2: 1/2 lIn

6

IX.1 2ds dp

pi,

18

A. The classical Plateau Problem for disc - type minimal .urfaces.

we can find

P as claimed.

o Proof of Lemma 4.3: Let X E c*(r), f > 0, Wo E fJB. We contend that there exists a number 6 > 0 depending only on f , D(X), the curve r, and the points Qi,1 $ j $ 3, such that for all wE fJB there holds

(4.2)

IX(w) - X(wo)1

< 2f ,

if

Iw - wol < 6.

This statement is equivalent to the contended equicontinuity of D-bc •. nded subsets of c*(r). By a theorem of Arzela - Ascoli the latter in turn is equivalent to the compactness of the injection C* (r) --+ CO( fJBj JR"). Choose

60

>

0 small enoug~ such that any ball of radius ~

v'6o

contains at most

one ofthe points Pi = e---r-, j = 1,2,3. Choose fa > 0 such that a ball ofradius fO in JR" contains at most one of the three points Qi' j = 1,2,3. Clearly, we may assume that f < fO. Choose fl, 0 < fl < f, such that for any two points X, Y on r at a distance IX - YI < fl there is a subarc I' c r with end-points X and Y contained in some ball of radius f in JR". (This is possible for any Jordan curve r. Otherwise, for sequences {Xm}, {Ym } of points in r with IXm - Yml --+ 0 (m --+ 0) any subarc I'm joining Xm with Ym would intersect fJBf(Xm ) in a point Zm. By compactness of r w· may assume Xm --+ X, Ym --+ Y = X, Zm --+ Z, IX - ZI = f. In particular, the limits Y and X correspond to different parameter values of a given parametrization of r. But this contradicts our assumption that r is a Jordan curve, i.e. a homeomorphic image of 8 1 .) By choice of fa, for X, Y E r with IX - YI < fl the subarc fer connecting X and Y and lying in a ball of radius f in JR" is unique and is characterized by the condition that f contains at most one the points Qi' j 1,2,3.

=

Now choose a maximal 6, 0

X 2 containing at most one of the points Qi,j 1,2,3. By monotonicity X(Cp ) f. Moreover, by Holder's inequality:

=

=

=

=

=

IX l

-

X 2 12

< ( / IX. Ids ) '

"

cp ::;; 811" D(X)/Iln

.p

! IX. I'd.

cp

pi ::;;

1611" D(X)/Iln 61 ::;; f~.

I. Exi.tence of a solution.

19

By choice of fll f' is con~ained in a ball of radius f. In particular, for any wE aB n B6(W o ) C Cp there holds

Since 6 depends only on D(X) and f, Qj, j 1,2,3, the proof is complete.

€,

=

f1

Wo

and

while the latter only depends on []

Lemma 4.3 immediately implies the following results: Proposition 4.5: in H i ,2(BjJRn).

The set

C* (r) is closed with respect to the weak topology

Proof: Consider a sequence {Xm} C C*(f) such that Xm ~ X weakly in H 1,2(BjJRn). By weak convergence, {Xm} is bounded and in particular for some c E JR uniformly in m. Lemma 4.3 now implies that (a subsequence) Xm on aB. Hence X E C*(f), and C*(r) is weakly H1,2-closed.

--+

X uniformly []

Together with coercivenes of D on C(r) (cp. Example 3.5) and weak lower semi-continuity of D on H1,2(Bj JRn) (cp. Example 3.4. ii)) Proposition 4.5 implies: Proposition 4.6: Suppose f is a Jordan curve in JRn such that C(f);i:0. Then there exists a solution to Plateau's problem (1.1) - (1.3) parametrizing a minimal surface of disc-type spanning f. Proof: Indeed, let C* (r) be defined as above with reference to a conveniently chosen triple (Q1, Q2, Q3) of points on f. Theorem 3.3 guarantees the existence of a surface X E C* (f) such that

D(X)

= XEC*(I') inf D(X).

Moreover, for any X E C(f) by Lemma 4.2 there is a unique conformal diffeomorX 0 9 E C*(f). By conformal invariance of phism 9 of B such that X' D also D(X/) D(X), and it follows that

=

=

inf XEC*(I')

D(X)

= XEC(I') inf D(X).

Consequently, X. minimizes Dover C(f). Hence by Remark 2.5.iii) X furnishes a solution to the parametric form (1.1) - (1.3) of Plateau's problem. []

20

A. The classical Plateau Problem for di.c - type minimal.urfacea.

For later reference we also note the following compactness result: Proposition 4.7:

r

Suppose

m the set

is a Jordan curve in II('. Then for any

f3 E

{X E c*(r) I D(X) $ f3}

is compact with respect to the weak topology in H1,2(BiJR"') and the CO(oBi JR"') -topology of uniform convergence on oB. Proposition 4.7 is easily deduced from Proposition 4.5 using the coerciveness and weak lower semi-continuity of D on C*(r). Combining Proposition 4.7 and Theorem 3.1 would give an alternative proof of Proposition 4.6.

It remains to give a general condition for in

Lemma 4.8:

C(r) to be non-void. This is contained

For any rectifiable Jordan curve r c II(' the class C(r)#0.

Our proof rests on the following a-priori bound for the area of solutions to (1.1) (1.3): Theorem 4.9 ( Isoperimetric inequality): Suppose r is a rectifiable Jordan curve in II(' with length L(r) < 00. Then for any solution X E C(r) of (1.1) - (1.3) there holds the estimate

4?1'D(X) $ (L(r»2. The constant 4?1' is best possible.

Cpo Nitsche [1, §323]. For our purposes it will be sufficient to establish the qualitative bound

D(X) $ c(L(r»2

(4.3)

for any C1(Bi JR"')-solution to (1.1) - (1.3). Proof of (4.3):

Multiply (1.1) by X and integrate by parts to obtain

2D(X)

= !IVXI 2 dW = B

!o",X. X do 8B

$!lo",XIIXldO $

IIrilLoo !lorXldo,

8B

8B

where do denotes the one-dimensional measure on oB, and normal and tangent vector fields to oB. Of course, by (1.3)

!

8B

18r XIdo

= L(f),

nand

T'

are unit

21

I. Existence of a solution.

while by suitable choice of coordinates in II(' such that

This proves (4.3) with c

0E

r

= 1.

o

Proof of Lemma 4.8: Approximate r by smooth Jordan curves r m in JRnf2 of class C 2 on BB=IR/21f. This can be done as follows: Let '1 E Hl.l(BBjJR) be a homeomorphism '1: BB -+ r. First convolute '1 with a sequence {'Tm} of non-negative 'Tm E COO(IR) vanishing for 14>1;::: ~ and satisfying J'Tm (4))d4> 1 to obtain a sequence of smooth maps 'Ym(4)) ==

=

.f1.

J'Y(4) - 4>') 'Tm (4)')d4>'. Then let 'Ym(4)) = ("Ym(4», ~ei 0, (4.4) and the proof of Lemma 4.3 show that the surfaces Xm are equicontinuous on BB. Hence a subsequence Xm ~ X weakly in Hl.2(Bj IRn+2) and

4

uniformly on BB. Note that X is harmonic with X(BB) the maximum principle X E H 1 .2(Bj IR") and X E C(r).

= r c IRn.

Thus by

o Lemma 4.8 and Proposition 4.6 finally yield the following existence result of Douglas [1] and Rad6 [1]:

.Theorem 4.10: Let r be a rectifiable Jordan curve in II('. Then there exists a solution X. to (1.1) - ( 1.3) characterized by the condition

D(X)

= XEC(I') inf D(X) < 00,

and X parametrizes a disc-type minimal surface of least area spanning f.

22

A. The classical Plateau Problem for disc - type minimal surfaces.

5. Regularity. The preceding considerations establish the existence of a solution to the parametric Plateau problem (1.1) - (1.3). In order to interpret this solution geometrically we now derive further regularity properties. Note that the regularity question is two-fold: First we analyze the regularity of the parametrization; then we turn to the question whether the parametrized surface is regular enough to be admitted as a solution to Plateau's problem, i.e. whether it is embedded (or at least locally immersed). While the first question is completely solved by Hildebrandt's regularity result [1] , for the second question a satisfactory answer can only be given in case n = 3 which corresponds to the physical case. In this case the results of Osserman [1] , Alt [1], Gulliver [1], Gulliver, Osserman, and Royden [1], Gulliver and Lesley [1], Sasaki [1] and Nitsche [1, p. 346] show that the solutions of Douglas and Rad6 will be free of interior branch points and hence will be immersed over B-and even be immersed over B if r is analytic or has total curvature ~ 411'. For extreme curves, i.e. curves on the boundary of a region n c JR3 which is convex or more generally whose boundary has non-negative mean curvature with respect to the interior normal, Meeks and Yau [1] even have proved that a least area solution to (1.1) - (1.3) parametrizes an embedded minimal disc. Related results were obtained independently by Tomi and Tromba [1] ,resp. Almgren and Simon [1]. This extends an old result of Rad6 [2] for curves having a single valued parallel projection onto a convex planar curve. Simple examples show that without such additional geometric conditions on r in general least-area solutions to (1.1) - (1.3) need not be embedded. Below we briefly stirvey some of the most significant contributions to the regularity problem for parametric minimal surfaces and sketch some of the underlying ideas involved. Let us begin by recalling the fundamental regularity result of Hildebrandt [1] :

Theorem 5.1: Suppose r is a Jordan curve in IJtl, parametrized by a map 'Y E em,a(BB;JRn),m ~ 1, 0 < 0: < 1, which is a diffeomophism of BB onto r. Then any solution X E C(r) to (1.1) - (1.3) belongs to the class em,a(B; JRn). Moreover, if solutions are normalized by a three-point-condition, the em,a-norms of solutions X E C·(r) to (1.1) - (1.3) are uniformly a-priori bounded.

Hildebrandt originally required m ~ 4 ; the improvement to m ~ 1 is due to J.C.C. Nitsche [2] . An overview of the different proofs of the result is given in Nitsche [1.p. 283 ff.] Hildebrandt's approach is rather interesting in as much as it reveals the complexity hidden in the seemingly harmless equations (1.1) - (1.3). His basic idea is to reduce the boundary regularity problem for (1.1) - (1.3) to an interior regularity problem for an elliptic system by means of the following transformation: Suppose 'Y E em,a, m ~ 2. Let 'Y(w o) = Qo E r. There is a diffeomorphism W of class em,a of IRn such that W maps a normal neighborhood V of Qo on r to a normal neighborhood of 0 on the (new) Xl_ axis. Let

(5.1)

Y=WoX.

23

I. Existence of a solution.

By harmonicity of X, Y solves an elliptic system ~Y

(5.2)

= r(Y)(VY,

VY)

with a bounded bilinear form r, whose coefficients of class em- 2 •a depend continuously on Y. r corresponds to the Christoffel symbols of the metric gii (Y)

8 = 8yi w-1 (Y).

8 8yi

w-1 (Y,) 1:5 i, j :5 n.

By continuity, X-l(V) contains a neighborhood U of transformed surface Y thus satisfies the boundary conditions yi

=0

Wo

in

8B.

The

in U, i ~ 2,

while the conformality relations (1.2) and our choice of W give a weak form of the Neumann condition 8n yl = 0 in U. By refiection across 8B, the function Y hence may be extended as a solution to an elliptic system like (5.2) with quadratic growth in the gradient (5.3) in a full neighborhood of Wo E IR? The standard interior regularity theory (cf. in particular Ladyshenskaya - Ural'ceva [1, p. 417 f.p now enables us to bound the second derivatives of Y -and hence of X - in L in a neighborhood of Wo in terms of the Dirichlet integral of X and its modulus of continuity. By Theorem 4.9 and Proposition 4.7 both these quantities are uniformly bounded for any solution X of (1.1) -(1.3) which is normalized by a three-point-condition. In view of Sobolev's embedding theorem an H2.2-bound for X implies a bound for V X in LP, V P < 00. Returning to (5.3) the Calderon-Zygmund inequality yields that X E H 2 .P, V P < 00. In particular, V X E C a , Va < 1. The complete regularity now is a consequence of Schauder's estimates for elliptic equations (5.2), cf. e.g. Gilbarg - Trudinger [1, Theorem 6.30].

D In later chapters we will return to this aspect and actually see some of the techniques of elliptic regularity theory in performance, cpo Section 11.5.

Now we direct our attention to the regularity of the parametrized surface. Note that by the conformality relations (1.2) any solution X of (1.1)-(1.3) will be immersed in a neighborhood of points wEB where VX(w) t= 0 . Definition 5.2: A point wE B is called a branch point of X iff V X(w)

= o.

The behavior of X near a branch point can be analyzed by means of the following representation.

24

A. The classical Plateau Problem for disc - type minimal surfaces.

Recall that if X u + iv given by

is harmonic, the components of the function

(5.4')

F

are holomorphic over complex integration

B.

= (Xu

- iXv )

Conversely,

X

F

of

w

= 8X

may be reconstructed from

F

by

(5.4")

Moreover, conformality is equivalent to the relation

(5.5)

( componentwise complex multiplication).

F·F=O

An interior branch point now may be characterized as a zero of the holomorphic vector function F. Since zeros of holomorphic functions are isolated this is also true for interior branch points of minimal surfaces X. Moreover, if X can be analytically extended across a segment C of 8B X can have at most finitely many branch points on any compact subset of B U C. This observation leads to the following result of Douglas [1] and Rad6 [1]:

Theorem 5.3: If X E C(r) is a minimal surface bounded by a Jordan arc then X!8B: 8B --+ r is a homeomorphism.

r

Proof: It suffices to show that X!8B is injective. Assume by contradiction that X(wt} X(W2) for Wd:W2 E 8B. Since X maps 8B monotonically onto r it follows that X(w) == X(Wl) for wE C, where C C 8B is an open segment with end-points Wb W2' We may assume X(Wl) O. Extending X by odd reflection across C we obtain a surface

=

=

A

X(w)=

{X(W), _X(w)

which is harmonic in a neighborhood function

Moreover, C so that X == const. the claim.

F =F

j;j2"

N of

wEB wdB l"

C

gi ving rise to a holomorphic

X is conformal on N. But X == 0 on F must vanish identically in N. Hence also = O. In particular, X == 0 and X ~ C(r). The contradiction proves on

B so that also

F == 0 on C and

D Definition 5.4: as a zero of F. Let

Wo

The order of a branch point

W

of a surface

X

is its order

E B be a zero of F of m-th order. Then after'a rotation of coordinates

25

I. Existence of a solution.

where a = (al, ... an ) E Cl:'n satisfies:

JR 3 a l = ia 2 > 0, a 3 = ... = an = 0, as a consequence of (5.5). Hence if

x (wo) = 0 (Xl

(5.6)

, X has the expansion

+ iX2)(w) =c(w _

wo)m+l + O(lw - wolm+2) Xi(w) =O(lw - wolm+2), j ~ 3,

in power series of w - woo An analoguous formula of course holds for Wo E aB, if X is analytic in a neighborhood of Wo in B. Using results of Hartman-Winter [I] it is possible to give similar expansions for X near branch points on aB in general provided r is of class C 2 or C l •l , cpo Nitsche [1, §381] for references. As a particular consequence of (5.6) we immediately deduce the following

Theorem 5.5: Suppose r is a Jordan curve of class C l .!, and let X E C(r) be a solution to Plateau's problem (1.1) - (1.3). Then X has at most finitely many branch points. Moreover, the tangent plane to the surface continuously near any branch point.

X

behaves

Let us now specialize formula (5.6) to the case n = 3. There exist numbers a E JR, a > 0 ,b E CI:', b#O, I ~ 2 such that in powers of w - Wo :

(Xl + iX2)(W) = a(w - wo)m+l + O(lw - wolm+2) X 3 (w) = Real (b(w - wo)m+l) + O(lw - wolm+l+l). I.e. locally, X looks like an (m + I)-sheeted surface over its tangent plane through X ( w o ). These sheets need not all be distinct, e.g if the power series expansions for Xl, ... , X 3 only contain powers of (w - wo)k for some k ~ 2.

Definition 5.6: A branch point Wo of a surface X is called a false branch point if there exists a neighborhood U of Wo and a conformal mapping g: U -+ B, g(wo) wo , g # {id}, such that X 0 g X near woo

=

=

Otherwise Wo is called a true branch point of X. The following result of Gulliver, Osserman and Royden [I] - cpo also Steffen-Wente [ 1, Theorem 7.2 ] - excludes false (interior) branch points for minimal surfaces satisfying the Plateau boundary condition:

Theorem 5.7: Suppose r is a rectificiable Jordan curve in JRn , n ~ 3. Then a minimal surface X E C(r) cannot have false interior branch points. This result makes crucial use of Theorem 5.3. For solutions to (1.1) - (1.3) of least area in ~ also true branch points can be excluded by means of the following argument due to Osserman [1].His results were completed and extended by Alt [I], Gulliver [I], Gulliver-Lesley [I] .

26

A. The classical Plateau Problem for disc - type minimal surfaces.

Theorem 5.8: Suppose X E C(r) minimizes D in C(r). Then not have true interior branch points. If in addition r is analytic then not have true boundary branch points, either.

X does X does

The proo/uses the fact that near interior branch points Wo by (5.6) different sheets of X must meet transversally along a branch line through Wo; cpo Chen [1] . This allows to construct a comparison surface of less area by a cutting-and-pastingand-smoothing argument. Suppose for simplicity that z has a branch point at W o , and let 'Y = 'Yl U 'Y2 :

= (0, ft), 'Y2(t) = (0, -ft),

'Yl(t)

01: of orders 111:, 1 :::; k :::; q. Let K denote the Gaussian curvature of X. Then there holds the relation p q 1 / 1 1 + " Ai + "111: + IKI do :::; -/C(r). ~ L...J 211" 211" 3=1

1:=1

X

In particular, if /C(r):::; 411" any minimal surface spanning r is immersed. Remark: Note that in case /C(r) = 411" a branched minimal surface X would have to satisfy K == 0, i.e. be a planar surface. Hence X could not have a branch point by the Riemann mapping theorem. The following example from Nitsche [1, §288] illustrates that in general even areaminimizing parametric solutions to the Plateau problem (1.1) - (1.3) may fail to be embedded (and hence will be physically unstable):

28

A. The cl....sical Plateau Problem for disc - type minimal surfaces.

For curves like the depicted one the true physical solutions apparently can be described best by the methods of geometric measure theory, cf. Almgren [1]. However, there is a class of curves in IFf where the least-area solution to the parametric Plateau problem can be shown to be a minimal embedding: these are the so-called extreme curves, i.e. Jordan curves on the boundaries of convex regions n E IR3. More generally, one can also allow curves on boundaries of regions n with the property that the mean curvarure of an with respect to the interior normal is non-negative (" M -convex" regions). Convex or M-convex surfaces provide natural "barriers" for minimal surfaces (by the maximum principle for the non-parametric minimal surface eqution, cf. Nitsche [1, §579 1£.]). The following result is due to Meeks and Yau[l]j the existence of an embedded minimal disc was established independently by Tomi and Tromba [1] , resp. Almgren and Simon [1] : Theorem 5.10: Let n let r c an be a rectifiable exists an embedded minimal (1.1)-(1.3) with X(B) C n

be an M-convex region in IR3 of class C 2 and Jordan curve, which is contractible in O. Then there disc with boundary r. Moreover, any solution X of which minimizes D in this class is embedded.

As a special case Theorem 5.10 contains the "existence" part ofthe following classical result of Rad6 [2]. The uniqueness is a consequence of the maximum principle. Theorem 5.11: Suppose r is a Jordan curve in IR3 having a single-valued parallel projection onto a convex curve in some plane P in JRl. Then there exist a unique minimal surface spanning r (up to conformal reparametrization). This surface is a graph over the region bounded by in P.

r

r

In higher dimensions a result like Theorem 5.10 is not known.

29

I. Existence of a solution.

Appendix We establish (2.5). For simplicity we assume in addition that Proposition A.l:

Let r be a inf

XEC(r)

G2

Jordan curve. Then C(r)#0 ,and

-

D(X)

r E G2•

= XEC(r) inf A(X).

Proof: Let X E C(r) n G 2(B; JRn). Approximate X by embedded surfaces Xe(u,v) == (X(u,v), eu, ev)..§ G2(B;JRn+2). We claim that for any e > 0 there

exists a map 9 E H i ,2 n CO(B; JR2) of B onto itself mapping BB monotonically onto BB and such that

D(Xe 0 g)

(A.I)

= A(Xe).

Since for any e > 0, and any such 9 there holds

X

0

9 E C(r), D(X 0 g) ::; D(Xe 0 g),

while as e --+ 0 clearly A(Xe)

--+

A(X) ,we infer from (A.I) that

D(X)::;

inf XEC(r)

By density of G 2(B, JRn) in implies the Proposition.

inf

XEC(r)nc 2(Bj.fl n )

A(X).

C(n and continuity of

A the latter inequality

In order to prove (A.I) introduce the set :F as the weak closure in H i ,2(B; JR2) of the set

9 is a diffeomorphism onto -B, 9 (21!'ik) e-r = e ¥

, k = 1,2,3 }

of normalized diffeomorphisms of B. Note Lemma A.2: :F is weakly closed in H i ,2(B; JR2). For any Z E c*(r) n G 2(B; JRn) we have Z 0 9 E c*(r), A(Z 0 g) = A(Z). Proof:

If g! !£. gk (m

--+

00), gk !£. g(k

--+

9 E :F, any

00) weakly in Hll2(B;JR 2)

where g! E:F, since H i ,2(B; JR2) is separable, a diagonal sequence g!.(k) !£. g, and :F is weakly closed. Next we show that any g E:F is in fact a uniform limit of diffeomorphisms. Indeed, a standard argument based on the Courant - Lebesgue Lemma 4.4 shows that H i ,2-bounded subsets of :F are equi-continuous: First, remark that :F c C*(aB), continuous on aB, cpo Lemma 4.3.

whence bounded subsets of

:F

are equi-

30

A. The classical Plateau Problem for elise - type minimal surfaces.

Next, given

K

>

:F(g) ~ K, any oBp(wo) n B :

>

0, I-'

0 there exists

II > 0 such that for any g E:F with p E [II, fol such that on Cp =

Wo E B there is a radius

sup Ig(w) - g(w'W 'ID ,",'EC p

~ 21fP j

lo,gl2ds

~ cIDI(g)1 < 1-'2. nil

Cp

Since g is a diffeomorphism and since the set {g E :F I D(g) ~ K} is equicontinuous on oB, for small I-' > 0 any such g maps the disc Bp(wo)nB onto the "small" disc bounded by g(Cp ). Hence sup Ig(w) - g(w')1 1'ID-'lD'I'}'EI be a Lipschitz continuous partition of unity subordinate to {V(:c.)}.o, i.e a collection of Lipschitz continuous functions 1/>. with support in V(:c.) such that 0:5 1/>. :5 1 for each L E 1 and

L 1/>.' (:c)

•'EI

= 1,

V:c E M .

35

II. Unstable minimal surfaces

E. g. we may let

p,(x)

t/I, (x)

= yEM\V(z,) inf Ix - yl p,(x)

- I>,,(x)· ,'EI

Finally, define

e(x)

= I: t/I,(x)(y, -

x),

'EI

where y, E M is associated to satisfies i) and ii) of Definition 1.7

x, by (1.1).

e

is Lipschitz continuous and

o E satisfies the Palau-Smale conditio,n on M if the following

Definition 1.7: holds: (P.S.)

Any sequence {x m } in M such that IE(xm)1 $ c uniformly, while g(x m ) --> (m --> 00) is relatively compact.

°

Remark 1.8: Again (P.S.) reduces to a variant of the well-known Palais-Smale condition (C), cf. Palais - Smale [1], in case M = T.

The Palais-Smale condition crucially enters in the following fundamental "deformation lemma." For 13 E JR let Mf3 = {x E M I E(x) < f3},

Kf3 = {x E MIE(x) =

13, g(x) = a}.

Lemma 1.9: Suppose E satisfies (P.S.) on M. Let 13 E JR, l > 0, and Suppose N is a neighborhood of Kf3 in M. Then there exists a number E E]O, l[ and a continuous one-parameter family ~ : [0,1] x M --> M of continuous maps q;(t,.) of M having the properties

i)

ii) iii)

q;(t,x) = x ift=O,

orifIE(x)-f3l2: l,

or ifg(x) = 0.

E(q;(t, x)) is non-increasing in t for any x,

q;(1, M f3 +E\N) C M f3 - E, q;(1, M f3 +E) C Mf3-E U N.

For the proof we need the following auxiliary ..Lemma 1.10: families

Suppose

Nf3,6

E satisfies (P.S.) on

= {z E M

M. Then for any

IIE(x) - 131 < 6, g(z) < 6}, 6> 0

13 E IR the

36

A. The classical Plateau Problem for disc - type minimal surfaces.

resp. Uf3,p

= {z E Mllz -

yl

< P for some

y E Kf3},p

>0

constitute fundamental systems of neighborhoods of Kf3.

Proof: By continuity of g, clearly each Nf3,6 and each Uf3,p is a neighborhood of Kf3. Hence it remains to show that any neighborhood N of Kf3 contains at least one of the sets N f3 ,6, Uf3 ,p. Suppose by contradiction that for some neighborhood N of Kf3 and any o > 0 we have N f3 ,6 q. N. Then for a sequence Om -+ 0 there exist elements Zm E Nf3,6m \N. By (P.S.) the sequence {zm} accumulates at a critical point Z E Kf3. Hence Zm E N for large m, a contradiction. Similarly, if Uf3,Pm rt. N for Pm -+ 0 there exist sequences Zm E Uf3,Pm \N, Ym E Kf3 such that IZm - Yml ~ 2pm. By (P.S.) {ym} accumulates at Y E K f3 , hence also {zm} does. A contradiction.

o Proof of Lemma 1.9: Coose numbers 0

< 0 < 0'

~

1, 0

E(:z:o), Choose E E]O, EO[ and let {:z: E MII:z: - :z:ol = E},

>0

such that

Y:z: E M, 0 < I:z: - :z:ol

< 2Eo.

{:Z:m} be a minimizing sequence for

E(:Z:m)

--+

inf

E in

Se(:z:o)

E(:z:) =: f3.

:r:ESe(zO)

If

f3 > E(:z:o) the proof is complete. Otherwise E(:Z:m) --+ E(:z:o) and either --+ 0 or there exists 00 > 0 such that g(:Z:m) ~ 00 for all m.

g(:Z:m)

In the first case, by (P.S.) {:Z:m} accumulates at an element :z: E Se(:z:o) , where E(:z:) = E(:z:o). Since this contradicts the strict minimality of :Z:o, we are left with the second case. Choose N = Nf3 ,60 ::::> Uf3 ,p ::::> Uf3 ,p/2 ::::> Nf3 ,6' ::::> Nf3 ,6 and let ip, '7, e, be defined as in Lemma 1.9. Consider the sequence Ym:= ( :Z:m). Since lei:::; 1 it follows that

-i,

E

3E

0< -2 -< IYm - :z:ol -< -2 < 2Eo· Moreover, since f3 = E(:z:o) :::; E((t, :Z:m» :::; E(:Z:m) :::; f3 t :::; ~, like (1.4) we obtain

E(Ym) =E(:Z:m) :::;E(:Z:m) -

+E

for large

m and

~ 021{t E [0, ~lI (t,:z:) rt Nf3 ,6'}1 • E P 2"1 02 . nun{2" , 2"}.

But since E(:Z:m) --+ E(:z:o) (m --+ 00) this implies that m. The contradiction proves the lemma.

E(Ym) < E(:z:o) for large

o Lemmata 1.9,1.11 immediately yield the following variant of the classical "mountainpass-lemma" : Theorem 1.12: Suppose E satisfies (P.S.) on M, and let :Z:1,:Z:2 be distinct strict relative minima of E. Then E possesses a third critical point :Z:3 distinct from :Z:1I :Z:2. :Z:3 is characterized by the minimax-principle

E(:Z:3) = inf supE(:z:) =: f3

(1.5')

pEP zEp

where

(1.5")

P

= {p EC([O, 1];

M) I p(O)

= :Z:1, p(1) = Z2}.

39

II. Unstable minimal surfaces

Moreover

and

Z3

is unstable in the sense that

Z3

is not a relative minimum of E.

Proof: i) By Lemma 1.11 13 > sup{E(zd, E(Z2)}. Suppose by contradiction that 13 is a regular value of E, i.e. K{:I O. Choose N 0, f 1 and let E > 0, ~ be as constructed in Lemma 1.9.

=

By definition of

13

=

there is pEP such that sup E(z) xEp

Applying the map P. while by iii)

=

~(1,.)

< 13 + E.

p by property i) of

to

sup E(z)

< 13 -

~

the path

p'

= ~(I,p) E

E.

zEp'

The contradiction shows that

13

is critical.

Now suppose that E possesses only critical points of energy 13 which are relative minimizers of E in M. The set K{:I will then be both open and closed in M{:I = {z E MJE(z) :$ f3}; hence there exists a neighborhood N of K{:I in M such that N and M{:I\K{:I are disjoint. A fortiori, then M{:I_E and N will be disconnected for any E > O. Choosing E > 0, ~ corresponding to this Nand f 1, and letting pC M{:I+€, however by property iii) of ~ we obtain a path

ii)

=

p'

= ~(I,p) E P,

p' C M{:I_E U N.

Since p':3 Zl ~ N and since M{:I_€ and N are disconnected, p' C M{:I_E' The contradiction shows that E has an unstable critical point of energy 13.

o A slight variant of the preceding result is given in

..Theorem 1.13: Suppose E satisfies (P.S.), and let Z1, Z2 be two (not neccessarily strict) relative minima of E. Then either E(Z1) = E(Z2) = 130 and :1:1, :1:2 are connected in any neighborhood of K{:Io' or there exists an unstable critical point Z3 of E characterized by the minimax-principle (1.5). Proof:

Let

13

be given by (1.5). If K{:I consists only of relative minimizers of

E as in part ii) of the proof of Theorem 1.12 we deduce that for any sufficiently small neighborhood N of K{:I there holds N n M{:I_E = 0 for any E > O. Letting E > 0, ~ be as constructed in Lemma 1.9 corresponding to N, f = 1, and choosing PEP such that pC M{:IH, we obtain a path p' ~(l,p) E P connecting

=

40

A. The classical Plateau Problem for disc - type minimal surfaces.

ZI with Z2 in M{3-f U N. Hence p' C N and ZI and Z2 both belong to the same connected component of Kfj, in particular E(zI) E(Z2) {3.

=

=

o Along the same lines numerous other existence results for unstable critical points can be given. For our purpose, however, Theorems 1.12, 1.13 will suffice and we refer the interested reader to Palais [3) or Ambrosetti - Rabinowitz [1). Theorem 1.13 is related to a result by Pucci and Serrin [1) .

41

II. Unstable minimal surfaces

2. The mountain-pass-Iemma for minimal surfaces. In order to convey the preceding results to the Plateau problem we reformulate the variational problem in a more convenient way. At first we closely follow Douglas' original approach to the Plateau problem. Let 'Y: BB --+ r be a reference parametrization of the Jordan curve assume that 'Y is a homeomorphism.

r.

We

Note that by (1.1) it suffices to consider surfaces X whose coordinate functions are harmonic: Co(r) = {X E C(r)16X = O}. By composition with 'Y and harmonic extension Co(r) may be represented by the space of monotone reparametrizations of BB=IR/27r. More precisely, let defined by

h: CO(BB)

--+

CO(B)

be the harmonic extension operator

6h( 0 z.

E {id}

+ T.

o Remark 2.4: The proof of Lemma 2.3 implies that the mapping X given by (2.1) between M with the topology induced from T and Co(r) is bounded. Moreover, for any uniformly convergent sequence {zm} ofparametrizations Zm E M the constant ¢"'m given in (2.7) is uniformly bounded away from 0 . Hence if E(zm) ~ C < 00 uniformly, also IZml1/2 will be uniformly bounded. From now on we shall always endow the set M with the topology induced by the inclusion Me {id}+T. Similarly, Co(r) will be endowed with the H i •2 n LOO-topology. Lemma 2.5: The map X: M -+ Co(r) given by (2.1) extends to a differentiable map of the affine Banach space {id} + T into H i •2 n LOO(B; JRn) of class C"-1, if 'Y E C'", r ~ 2. Proof: Note that for any X E H i •2 n LOO(B; JRn) there is a unique harmonic surface XoEHi.2nLOO(B,lRn) which agrees with X on 8B in the sense that Xo E X + H;·2(B; JRn). Indeed, Xo is characterized by the variational principle

Existence of Xo follows easily from Theorem 1.3.2; necessarily Xo is harmonic; bounded ness and uniqueness are a consequence of the maximum principle. Now "define" the trace space Hl/2 .2(8B; JRn)=H i •2(B; JRn)/ H;.2(B; IRn) to be the set of equivalence classes XIOB=X + H;·2(B; JRn) endowed with the quotient topology. In particular, let

bethesemi-normon H l / 2 •2(8B;JRn). Infact H l / 2 •2(8B;JRn) is a Hilbert space with respect to the scalar product induced by L2(8B;JRn ) and the bilinear map

(XIOB,YloB)1/2

= (Xo,

Yoh= jVXoVYodW, B

where Xo and Yo are the unique harmonic extensions of XOB, YIOB resp. By construction, the harmonic extension h is a linear isomorphism from Hl/2 .2 n

45

II. Unstable minimal surfaces

L""(BBj JR") into H 1,2 n L""(Bj JR"). Moreover, by (2.4) an intrinsic definition of the semi-norm I· 11/2 in Hl/2,2 can be given in terms of Douglas' integral

D(X)

=

_1_!! IX(e' ) 2,.. 2,...'"

1611"

o

.",'

X(e: )1 2 d

(-I)',

(3.4 ')

,

m=O

L:(-I)mCm

= 1.

m=O

Remark 3.7: tity

The Morse relations (3.4') may be summarized in the single iden00

L: Cmtm = 1

(3.4 ")

+ (1 + t)Q(t)

m=O

where Q is a polynominal with non-negative integer coefficients, and the polynom00

inal

E

Rmtm ==

1 is the Poincare polynominal of M, cf. Rybakowski - Zehnder

m=O

[1, p.124], the numbers

Rm =

I, ( ( )) rank Hm M = { 0,

m

m

=0 >0

denoting the Betti numbers of the (convex hence) contractible space M . ..Proof: By (3.2) and our non-degeneracy assumption critical points are isolated. By (P.S.) the set of critical points having uniformly bounded energy is compact, hence finite if it consists of isolated points. This proves i) and the first part of iii). Postponing the proof of ii) for a moment let us derive the Morse inequalities (3.4). Let /31 < ... < /3j be the critical values of E and choose regular values ai, 'Yi such that a1 < f31 < 'Yl = a2 < f32 < ... f3i < 'Yi' , For each pair of regular values a, 'Y let

56

A. The classical Plateau Problem for disc - type minimal surfaces.

be the Betti numbers of the pair (M..,., Ma ), and let

e::.,"" = I{:c EM..,. \Mal g(:c) = 0,

Index(:c)

= m}l.

By ii) for each pair ai,1'i M""i is homotopically equivalent to Mai with ki handles of types ri, ... disjointly attached,. where rL ... , r~. are the Morse indices of the critical points of E at energy f3i. By Remark 3.5. ii) therefore the Betti numbers of (M""i' M ai ) are the same as those of a disjoint union of ki pointed spheres (Sd,p) of dimensions d = ri, ... , r~ ..

.

.

ri.



Since

{~

,m=d , else,

we obtain the relations

Adding, cycles may cancel while critical points cannot and we obtain (cf. Palais [1, p. 336 ff.]) the system of inequalities for all regular a < l' :

, ~ e::.,"", V m E,IN L (_I)'-m R!"" ~ L (_I)'-me::.,"",

R!"" (3.5)

0

m=O

V I E INo

m=O

00

00

m=O

m=O

Equality in the last line corresponds to the well-known additivity of the Euler characteristic. Letting a --+ -00, l' --+ 00 the right hand sides of (3.5) stabilize for large a, l' while the quantities on the left for large a, l' are bounded from below by the corresponding expressions involving the Betti numbers Rm of M. This completes the proof of (3.4). It remains to establish ii).

Preliminaries: Let a < l' be regular values of E and for simplicity assume that :Co E M is the only critical point of E in M having E(:c o ) = f3 E [a, 1']. Let robe the index of :Co, H = H + ffi H _ the standard decomposition of H at :Co. For any {E H denote {= {+ + {_ E H+ ffi H_ its components. Choose 0 < p < 1 such that (3.6) which is possible by assumption (3.3). By non-degeneracy of :Co there is a constant .A > 0 such that (3.7 ')

57

II. Unstable minimal. surfaces

By (3.2) we may suppose that

p is chosen such that

(3.7 ") for all z E B 2p (zo, H) n M, all y E M, provided particular, the vector field eo given by

is a pseudo-gradient vector field for sense: By (3.6) for any x E U

E on

Iz - ylH

~

Iz - zolH. In

U:= Bp(x o , H) n M in the following

eo(x) + x = 2(x - x o)_ - (z - x o) + x = Zo

+ 2(z -

zo)_ EM.

Moreover, while by (3.7)

g(z)

~ c·

(dE(x), x - y)

sup "EM

la-"IH 2, and let Zo E Mt be a non-degenerate critical point of E on Mt . Then Zo is a C 2 -diffeomorphism of the interval [0,211"] onto itself.

eE

Proof: A boundary branch point gives rise to a "forced Jacobi field" ker d2 E(zo) C Ht ; cpo Bohme-Tromba [1 ,Appendix I] .

o By Lemmata 4.4, 4.5 now also assumption (3.3) will be satisfied. The Palais - Smale condition is a consequence of Lemma 2.10. We summarize:

Theorem 4.6: Suppose 'Y E cr(8B; llln), r ~ 5, is a diffeomorphism onto a Jordan curve r, and assume that r bounds only minimal surfaces Xo X(zo) whose normalized parametrizations Zo E Mt correspond to non-degenerate critical points of E on Mt C id + Tt C id + Ht in the sense of Definition 3.2. Then the Morse inequalities (3.4) hold.

=

Remark 4.7: Below we shall see that as a consequence of the "index theorem" of Bohme and Tromba [1] the non-degeneracy condition is fulfilled for almost every r in II{', n ~ 4, cf. Corollary 6.14. To give an "intrinsic" characterization of non-degeneracy of a minimal surface Xo X(zo) E Co(r) let us introduce the space

A

H=

=

{A 12 n A A 8 } eEH' (B;lll )1b.e=O, e is proportional to 8tj>'Y(zo)along 8B

of harmonic surfaces tangent to

Xo. Formally,

II

is the "tangent space" to

Co(r) at Xo in H 1 ,2(B;lll3). Note that since 'Y is a C 2 -diffeomorphism onto account of our regularity result) the linear map (4.2)

is an isomorphism between H and

fl.

Now compute the second variation of D on Co(r):

r

and since

Zo E C1 (on

64

A. The classical Plateau Problem for disc - type minimal.urfaces.

Let {,11 E H,

(= dX(zo)' {, 17 = dX(zo)'l1

d 2 D(Xo)(t, 17)

= d2 E(zo)({, 11) = 8n X o' d~2'Y(ZO)' {'11 do +

E

H. Then by (4.1)

J

J

DB

B

V(dX(zo) . {). V(dX(zo) . l1)dw

d2

J

8nXo . J4IXo ( d ) (d ) = 1d 12 1 d 12 d.p'Y(Zo)·{ . d.p'Y(zo)·l1 do DB ~'Y(Zo) . ~zo

+

J

V{V17dw.

B

But the expression

r

equals the geodesic curvature of formula may be simplified

d 2 D(Xo)(t, 17)

= XolDB

J J

DB

B

r

=

vtV17 dw -

B

(4.4)

=

vtV17 dw -

in the surface

J J

Xo. Hence the above

ICXo(r)t17·1 d~ Xol do

ICX o(r)t17 dr,

v t,

17 E H,

and we have obtained the following result of B5hme [1] and Tromba [1]: Proposition 4.8:

Suppose

r

E C", r

~

3. Then at a minimal surface

Xo E

C(r) the second variation of Dirichlet's integral on fI is given by (4.4). Remark 4.9: Since dX(zo): H -+ fI is an isomorphism it is immediate that the components of the standard decompositions

are mapped into one another under dX(zo). Moreover, dX commutes with the conformal group action. Hence Xo X(zo) will correspond to a non-degenerate critical point in Mt, iff Ho dX(zo)(TsdG) and the Morse index of Zo is given by dim H_ .

=

=

We close this section with a question posed by Tromba which is related to (4.4) and the following uniquenes result of Nitsche [3] : TheorelD 4.10: Suppose r c JR!3 is an analytic Jordan curve, and assume that the total curvature of r: IC(r) ::; 411". Then (up to conformal reparametrization) r bounds & unique minimal surface.

II. Unstable minimal surfaces

65

The proof uses the mountain pass lemma Theorem 1.12 and the fact that under the curvature bound ~(r)::; 411" any solution Xo = X(xo) to (1.1) -(1.3) is strictly stable in the sense that for some A > 0 : (4.5) for all (E dX(xo)(Ht). Is there a way of deriving (4.5) from (4.4) directly?

66

A. The classical Plateau Problem for elise - type minimal surfaces.

In this chapter we present the proofs of Proposition 2.10 and

5. Regularity. Lemma 4.4.

Propositon 5.1: Suppose that 'Y E C r , r;::: 3, and let point of E on M, satisfying the variational inequality

!

(5.1)

z E Mt be a critical

d~'Y(Z).(Z-Y)do5: °

8n X.

DB

X=X(z). Then XEH 2,2(BjJR.n ).

for all YEMt, where

For the proof of Proposition 5.1 we need to introduce difference quotients in angular direction: 1 8,.~(.p) == h[~(.p + h) - ~(.p)l, etc. and translates Note the product rule and the following formula for integrating by parts 2...

! o

*

*

2...

2...

2,..

8,.~1]d.p = ![~+1] - ~1]ld.p = ![~1]- - ~1]ld.p = - ! ~8_,.1] d2

d () . -'Y:r: d (") d:r: " d:r: , -'Y:r: d¢>

x

(Note that since l' is a diffeomorphism the denominator in this expression is uniformly bounded away from 0.) In consequence D(8h :r:) is bounded by the Dirichlet integral of the above right hand side:

f

IV8h:r:1 2dw

B

~c

f

IV:r:12 (18hX1 2 + 18h:r:1 2) dw

B

+ cll:r:+ - :r:llioo

f

IV8h:r:1 2 dw + c

B

and for

e and

IV8h Xl 2 dw,

B

h sufficiently small there results

J

J

B

B

(IV8h XI 2 + IV8h:r:12) dw ~ c

*

J

(IVXI2 + IV:r:12) (I8hXI 2 + 18h:r:1 2 ) dw.

Since 8 h z is 2?f-periodic we may regard

8hZ

as a function on 8B=1Rh?f'

69

II. Unstable minimal surfaces

In order to bound the products on the right two more auxiliary results are needed. The first lemma states the "self-reproducing character" of Money spaces, cpo Morrey [1, Lemma 5.4.1, p.144]: '" E HJ·2(B)

Lemma 5.3: Suppose growth condition

J

1/J

and

E Ll(B) satisfies the Morrey

11/Jldw ~ corll

Br(wo)nB for all r > 0, Wo E B with uniform constants Ll(B) and for all r > 0, Wo E B there holds

J

11/J",2Idw ~

C1Co

Co and

r ll / 2

Br(wo)nB with a uniform constant

J.I.

>

O. Then

1/J",2

E

JIV",12dw B

Cl.

The second auxiliary result establishes the Money growth condition for the functions

1/J

= IVXI 2+ IV:c1 2,

Lemma 5.4: Under the assumptions of Proposition 5.1 there exist constants Co, J.I. > 0 such that for all r > 0, Wo E B there holds

J

IVXI 2+ IV:c1 2dw ~ corll

Br(wo)nB

JIVXI2+ IV:c1 2dw. B

=

Proof: Fix Wo eitf>o E aB, and let :Co be the mean of :c over the "annulus" (B2r(W o )\Br(wo)) naB; also let r E Coo be a non-increasing function ofthe distance Iw - wol satisfying the conditions 0 ~ r ~ 1, r == 1 if Iw - wol ~ 2r, r == 0 if Iw - wol ~ 3r, IVrl ~ clr, IV2rl ~ clr2, Then for

3r

o.

By Lemma 5.2

f E H 2,2(Bj JRn).

Inserting fourth order difference quotients O_hOhO_hah~' in a similar manner for Xo E C 4 (Bj JR3) we obtain that f E H3,2«Bj JRn)) '-+ C1(Bj JRn ), and hence the claim.

o

78

A. The classical Plateau Problem for disc - type minimal surfaces.

6. Historical remarks. The solution of Plateau's problem fell into a period of very active research in variational problems. Only a few years before Jesse Douglas' and Tibor Rad6's work on minimal surfaces L. Ljusternik and L. Schnirelmann had developed powerful new variational methods which enabled them to establish the existence of 3 distinct closed geodesics on any compact surface of genus zero. Also in the 20's Marston Morse outlined the general concept of what is now known as Morse theory: A method for relating the number and types (minimum, saddle) of critical points of a functional to topological properties of the space over which the functional is defined. Quite naturally, Morse and his contemporaries were eager to apply this new theory to the Plateau problem. In the following we briefly survey the Morse theorical results obtained for the Plateau problem by Morse-Tompkins [1] and independently by Shiffman [1] in 1939. Necessarily, this account cannot accurately present all the details of these approaches. Nevertheless, I hope that I have faithfully portrayed the main ideas.

The work of Morse-Tompkins and Shiffman. Morse-Tompkins and Shiffman approach the Plateau problem in the frame set by Douglas. I.e. surfaces spanning r are represented as monotone maps x E M* of the interval [0,21r] onto itself, preserving the points (21rk)/3, k = 1,2,3 and their areas are expressed by the Dirichlet-Douglas integral E, cf. (2.3), (2.4). The non-compactness of the space M* is no problem. In fact, the principles that Morse had developed apply to any functional £ on any metric space (M, d) provided the conditions of "regularity at infinity", "weak upper-reducibility", and "bounded compactness" are satisfied. This latter condition is crucial. It requires that for any a E IR the set (6.1)

{x E MI £(x) ::; a}

is compact.

=

By Proposition 2.1, the functional £ E will satisfy the condition of bounded compactness on M = M* if we endow M* with the CO-topology of uniform con vergence. This choice of topology therefore is the natural choice that Morse-Tompkins and Shiffman take. However, in this topolgy E is only lower semi-continuous on M*, cf. Remark 1.3.2. For a functional which is not differentiable the notions of a critical point and its critical type are defined with reference to neighborhoods of a point Xo E M with £(x o ) = f3 in the level set M{J. Definition 6.1: Let f3 E JR, and let U C M{J be relatively open, 11': U x [0,1] --+ M a continuous deformation such that 11'(.,0) = idl u . Let Vee

79

II. Unstable minimalaurfacea

U. cp possesses a displacement function all z E V, 0::; 8 ::; t ::; 1 there holds e(cp(x,

s» -

6: IR+ U {o}

e(cp(z, t)) ~ 6 (d(cp(z,

and

6(e)

=0

iffe

-+

IR+ U {o} on

V iff for

8), cp(z, t)))

= O.

The deformation cp is an e - deformation on function on any Vee u.

U if cp possesses a displacement

=

Definition 6.2: Zo E M with e(zo) {3 is homotopically regular if there is a neighborhood U of Xo in Mf3 and an e-deformation cp on U which displaces Zo (in the sense that cp(zo,I)#zo). Otherwise, Zo is homotopically critical. Remark 6.3: Definitions 6.1, 6.2 imply that for a homotopically regular point Zo there exists a deformation cp: U x [0,1] -+ M of a neighborhood U of Zo in Mf3 such that

= x,

i)

cp(z, 0)

ii)

e( cp( x, t))

iii)

Vx E U,

is non-increasing in t, Vx E U,

For any Vee U there is a number

f>

0 such that

cp(V, 1) C Mf3-f'

By (6.1) at a regular value f3 finitely many such neighborhoods cover

{x E Mle(x)

= {3}.

Piecing deformations together we thus obtain a homotopy equivalence

for some

f

> 0, for any regular value {3, as in the differentiable case.

Examples 6.4: i) If Xo is a relative minimum of e on M then Zo is homotopically critical. Indeed, for suitable U:3 XO we have UnMf3 {x o }, and U cannot admit an e-deformation which displaces Xo'

=

ii)

Let M=IR 2 ,e(x,y)=z2_ y 2. The point (0,0) is homotopic ally critical since Mo is connected while for any f > 0 and any neighborhood U of (0,0) the set M-f n U is not.

=

iii) Let M IR, e(z) critical iff d is even.

= z d, dE IN.

The point

Xo

=0

is homotopically

80

A. The classical Plateau Problem for disc - type minimal surfaces.

Examples 6.4 illustrate that the concept of a homotopically critical point is natural but somewhat delicate. In general, in order to be able to decide whether for a differentiable functional £ E Cl(M) a critical point :Co E M (in the sense that d£(:c o ) = 0) is also homotopically critical one needs to analyze the topology of the level set MfJ near :Co. Unless :Co is a relative minimum, this analysis in general requires that £ E C 2 near :Co and that d 2 £(:c o ) is non-degenerate. For the Plateau problem we have the following result, Morse - Tompkins [1, Theorem 6.2]: Lemma 6.5:

with the spanning

c o_ r.

If:c o E M* is homotopically critical for E on M* endowed topology, then Xo = X(:c o) parametrizes a minimal surface

Information concerning the critical type of

:Co

is captured in the following

Definition 6.6: Let :Co E M be an isolated homotopic ally critical point of £ with £(:c o ) = /3, and let U C MfJ be a neighborhood of :Co containing no other homotopically critical point. Then

lim inf rank (Hk

(U,Ma))

a//3 is the k-th type number of

:Co.

The following observation is crucial: Lemma 6.1: Example 6.S.

The numbers td:c o) are independent of U.

i)

If

:Co

is a strict relative minimum of

£ on a metric

space M, then

k=O else

ii)

If:c o is a non-degenerate critical point of £ E C2(M) k = Index(:c o ) else

Unless :Co falls into the categories i), ii) of Example 6.8 in general it may be impossible to compute its type numbers. Now let

Rio

= rank (Hk (M)),

Tk =

~ tk(:C) '" hom. crit.

81

II. Unstable minimal surfaces

be the Betti numbers of M and type numbers of C, resp. Then Morse's theory asserts:

Theorem 6.9: Suppose C: M -+ IR satisfies the conditions of "regularity at infinity", "weak upper-reducibility", and "bounded compactness" and assume that C possesses only finitely many homotopic ally critical points. Then the inequalities hold: m

m

k=O

k=O 00

00

k=O

k=O

For the Plateau problem Theorem 6.9 has the following corollary, cpo MorseTompkins [I, Corollary 7.1], which is slightly weaker than our result Theorem 2.11:

Theorem 6.10: Suppose r bounds two distinct strict relative minima Xli X 2 of D. Then there exists an unstable minimal surface X3 spanning r, distinct from Xl, X 2 • Proof:

Note that since M* is contractible its Betti-numbers Rk

=

{I, 0,

k=O else

By Example 6.8, i)

To 2': 2, whence Theorem 6.9 for

m

=1

gives the relation

Hence E must possess a critical point :1:3 such that any neighborhood of :1:3 in M* contains points :I: with E(x) < E(:l:3). I.e. X3 X(:l:3) is an unstable minimal surface.

=

o But what is the relation of Theorem 6.9 in the case of the Plateau problem with our Theorem 4.6? Are these results equivalent - at least in case r spans only finitely many minimal surfaces which are non-degenerate in the sense of Definition C k in this case? The answer to 3.2? In particular, is it possible to identify Tk this question is unknown. In fact, the CO-topology seems too coarse to allow us to compute the homology of CO-neighborhoods of critical points of E in terms of the second variation of E near such points - even if we use the H1/2,2-expansion Lemma 4.2 . It is not even clear if such points will be homotopically critical points of E in the sense of Definition 6.2 and will register in Theorem 6.9 at all.

=

82

A. The classical Plateau Problem for disc - type minimal surfaces.

The technical complexity and the use of a sophisticated topological machinery (which is not shadowed in our presentation) moreover tend to make Morse-Tompkins' original paper unreadable and inaccessible for the non-specialist, cf. Hildebrandt [4, p. 324]. Confronting Morse-Tompkins' and Shiffman's approach with that given in Chapter 4 we see how much can be gained in simplicity and strength by merely replacing the CO-topology by the Hl/2 ,2-topology and verifying the Palais - Smale - type condition stated in Lemma 2.10. However, in 1964/65 when Palais and Smale introduced this condition in the calculus of variations it was not clear that it could be meaningful for analyzing the geometry of surfaces, cf. Hildebrandt [4, p. 323 f.]. Instead, a completely new approach was taken by Bohme and Tromba [1] to tackle the problem of understanding the global structure of the set of minimal surfaces spanning a wire.

83

II. Unstable minimal.urCace.

The Index Theorem of Bohme and Tromba and its consequences.

Bohme and Tromba turn around completely our view of the classical Plateau problem. If to this moment we have only looked at surfaces with a Jized boundary r, now Bohme and Tromba consider the bundle of all surfaces spanning any J ordan curve in IK' . If we had so far tried to understand the structure of minimal surfaces with given boundary, Bohme and Tromba analyze the structure of the set of all branched minimal surfaces in IR" . The information that we need in order to solve the Plateau problem for a given wire is contained in the properties of two differentiable maps: The (bundle) projection IT of a surface to its boundary, and the conformality operator K. Without going into technicalities we now present the main ideas of Bohme' and Tromba's approach. For details we refer the interested reader to the original paper of Bohme - Tromba [1] and to the papers by SchufHer - Tomi [1], Sollner [1], Thiel [1] ,[2] on extensions and simplifications of their approach.

Let

A be the space of diffeomorphisms 'Y: BB curves.

-+

IR"j this is the space of (parametrized)

Let D=

U

D"

"ENo

be the space of monotone parametrizations z of BB

o ~ 0, X = X 0 g-l E H 1,2 n LOO(iJ, JR3). Then V(X) =1/3/ Xu A Xv' Xdw fJ

(1.11)

=1/3/ Xu A Xv' X det

(d(g-l)

0

g)1 det(dg)ldw

= V(X).

B

v) If X E C(r) n C 2 (Bj JR3) is a stationary point of DH with respect to variations of the dependent and independent variables, cpo Lemma 1.2.2, from (1.10) and (1.11) we obtain the weak form of (1.1) (dDH(X),ip)

=/

VXVip + 2H Xu A Xv' ipdw

B

(1.12)

= / [-6X

+ 2H Xu AX.,]. ipdw =

0, Vip E C:,

B

resp. the conformality relations, cpo Lemma 1.2.4:

:€ DH (X 0 (id + (7)-1) I€=o = :€ D (X 0 (id + (7)-1) I€=o = 0,

(1.13)

I.e.,

X

1S

an

V7 E

c 1 (Bj JR2).

H -surface in conformal representation.

Remark 1.1. v) justifies our claim that the parametric H -surface problem (1.1)(1.3) formally corresponds to the Euler-Lagrange equations of DH on C(r). To make this precise we now analyze the volume functional

V in detail.

94

B. Surfaces of prescribed constant mean curvature

2. The volume functional. The basic tool in this section isoperimetric inequality for closed surfaces in IR?, cf. Rad6 [4).

IS

the following

Let X,YEH 1 ,2nL OO (BjIR?) satisfy X-YEH;,2(BjIR?).

Theorem 2.1: Then

3611" IV(X) - V(YW 5 [D(X)

+ D(yW,

and the constant 3611" is best possible.

Remark 2.2: = X_ where

The constant 3611" is achieved for example if X

i)

= X+,

Y

denote stereographic representations of an upper and a lower hemi-sphere of radius 1 centered at o. ii) Recall that V is invariant under orientation-preserving changes of parameters. Moreover, by the f-conformality Theorem 1.2.1 of Money we may introduce coordinates on X to achieve D(X) 5 (1 + f)A(X) for any given f > o. Hence Theorem 2.1 implies the estimate

3611"1V(X) - V(Y)12 5 [A(X)

+ A(YW

for all X, Y E H 1,2 n LOO(Bj JR3) with the property that there exists an oriented diffeomorphism g of B onto itself such that X!elD Y 0 g!elD.

=

Theorem 2.1 and Remark 1.1 have important consequences. The following result (like many results on the analytic properties of H -surfaces) is due to H.C. Wente

[1) . Theorem 2.3: i) For any X E H 1,2 n LOO(Bj JR3) to an analytic functional on X + H;,2(Bj JR3).

V continouslyextends

V has the expansion in direction ep E H;,2(Bj JR3) : (2.1) ii) (2.2)

V(X

+ ep) = V(X) + (dV(X), ep) + (1/2

)d 2V(X)(ep, ep)

+ V(ep).

The first variation dV given by

(dV(X), ep)

=

J

Xu 1\ X • . epdw, Vep E H;,2 n LOO(Bj JR3)

B

continuously extends to a map dV: Hl,2(Bj JR3) _ (H,!,2(Bj JR3») * which satisfies the estimate

l2.3)

!(dV(X), ep)! 5 cD(X) D(ep)1/2 ,

m. The

existence of surfaces of prescribed constant mean curvature

95

and is weakly continuous in the sense that

(2.4)

Xm ~ X in H 1,2(B; JR") => (V(Xm ), tp)

->

(dV(X), tp),

V tp E H;,2(B;JR3). iii) (2.5)

The second variation d2 V given by

d 2V(X)(tp, 'I/J)

=

J

(tpu /\ 'I/J'IJ

+ 'l/Ju /\ tp'IJ)' Xdw,

V tp, 'I/J E H;,2(B;JR3)

B

continuously extends to a map d 2V: H 1,2(B; JR3) satisfies the estimate

--+

(H;,2 x H;,2(B; JR3))* which

(2.6) and is weakly continuous in the sense that

(2.7)

Xm ~ X

in H 1,2(B;JR3)

=> d2V(Xm )(tp,'I/J)

--+

d 2V(X)(tp,'I/J),

V tp, 'I/J E H;,2(B;JR3). Moreover, d 2V(X) for fixed X E Hl,2(B; JR3) is a completely continuous bilinear form on H;,2(B; JR3) in the sense that

(2.8)

tpm ~ tp, 'l/Jm ~ 'I/J in H;,2(B;JR3)

==> d2 V(X)( tpm, 'l/Jm) iv)

If

d2 V(X)( tp, 'I/J).

Xm w ~ X·In

H 1,2(B', JR3) whl'le

V(Xm) --+ V(X) , (dV(Xm)' tpm) --+ (dV(X), tp) , d2 V(Xm )(tpm,'l/Jm) --+ d2 V(X)(tp,'I/J)

(2.9) (2.10) (2.11) as m

X m, X E C(r) an d

--+

--+ 00.

Proof: By (1.6), (1.10) formulas (2.1)' (2.2), (2.5) hold for X E C 2(B; JR3), tp, 'I/J E Cgo(B; JR3). By uniform continuity of the integrals J Xu /\ X'IJ . tpdw with B

respect to X E H 1,2(B; JR3), tp E H 1,2 n LOO(B; JR3) it is also clear that dV continuously extends to dV: H 1,2(B; JR3) --+ (H;,2 n LOO(B; JR3»*. Similarly, by (1.9) and (2.5) d 2 V extends to a map

Once we have established (2.3), ( 2.6) ,moreover, dV(X) extends to a continuous linear functional on H;,2(B; JR3) while d 2V(X) continuously extends to a bilinear form on [H;,2(BimJ)]2 as claimed.

96

B. Surfaces of prescribed constant mean curvature

(2.3) and (2.6) are deduced from Theorem 2.1 as follows:

X = (Xl,X2,X3) E H l ,2nLOO {BjJR 3) and OO L {BjJR3) let For

Y =

(

ip = (ipl,ip2,ip3) E H;,2 n

Xl X2 ) (Xl X2 ip3) D(X)1/2' D(X)1/2 ' Z = D{X)1/2' D{X)1/2 ' D{ip)1/2 .

°,

Note that V{Y) = 0, D{Y) :s 1, D{Z) Theorem 2.1 to Y and Z we obtain

:s 2.

IV{Z)1 2

Applying the isoperimetric inequality

:s 4~'

By antisymmetry of the volume element a /\ b . c now V is also trilinear in the components of Z = (Zl, Z2, Z3). Multiplying by D{X)2 D{ip) we hence find that IV(Xl,X2,ip3W l(dV{X),(0,0,ip3)}1 2 437r D{X)2D(ip).

:s

=

Repeating the argument for the remaining two components of ip (2.3) follows. To see (2.6) let

X, ip as above, 1jJ E H;,2 n LOO{Bj JR3), and set Xl

Y

)

= ( D(X)1/2' 0, ° , Z =

(Xl ip2 1jJ3) D(X)1/2' D(ip)1/2 ' D(1jJ)1/2 .

Then the above reasoning gives (denoting e.g.

Id 2V(X)(ip2, 1jJ3)12

(O, ip2, 0)

= ip2

for brevity)

= IV(Z)1 2D{X)D(ip)D(1jJ) :s ~!D(X)D(ip)D(1jJ),

and (2.6) follows by trilinearity of V:

Id 2V(X)(ip, 1jJ)1

:s L

Id 2V(X)( 1,

X(w),

X(w)

= { -x (

w )

we obtain a continuous weak solution X to (1.1) in JR2 with

D(XjJR2) = 2D(X) < Letting with

F(w)

= Xu -

iX" we hence obtain that

00.

F is a holomorphic function

°

By the mean value theorem F == and X is conformal. But then X can have only finitely many branch points on aB or X == const = 0. Since X == on aB, by conformality also VX == on aB, and the conclusion follows.

°

°

o

As in Struwe [2] and consistent with the remainder of this book Theorem 3.1 will now be deduced as an application of the Mountain - Pass - Lemma.in the following variant (cf. Theorem II.I.12):

122

B. Surface. of prescribed constant mean curvature.

Theorem 3.3: Let T be an (affine) Banach space, E E C 1 (T) and suppose E admits a relative minimum ~ and a point Z1 where E(Z1) < E(~).

Define

P = {p E Co ([0, Ijj T) I p(O)

(3.1)

=~,

P(1) = Zl}

and let {3= infsupE(z).

(3.2)

pEP II:Ep

Assume that E satisfies the Palais - Smale condition at level {3, i.e. the condition: Any sequence {zm} in T such that E(zm) -+ {3 while dE(zm) -+ 0 as m -+ 00 is relatively compact.

(P.S.)f3

Then E admits an unstable critical point

z

with E(z)

={3.

Proof of Theorem 3.3: Note that Lemma I1.1.10 with M = T remains true at the level {3 under the weaker compactness condition (P.S.)f3. But then also Lemma I1.1.9 remains true at level {3, and the proof of Theorem I1.1.12 conveys: For "l = E(~) - E(Z1) > 0 and any neighborhood N of the set Kfj of critical points Z of E with E( z) = {3 there exists a number f EjO,"l[ and a flow 0, and let

f, -00.

Assumption (4.2) turns Lemma 4.5 into a local compactness condition comparable to Lemma 3.5: (4.3)

+!lfr

For any /3 < /30 any bounded sequence Zm E M and DH(Xm ) --+ /3, gH(Zm) --+ 0 as compact.

/3

Define, as usual, for

Xm m --+

= X(zm) 00

with is relatively

E 1R

JC{3

= {z E M I EH(Z) = /3,gH(Z) = O},

M{3 ={z EM I EH(Z)

0 also introduce numbers

fjR Lemma 4.9 Suppose holds the estimate

= pEP in(

sup EH(:z:) ~ fj . -Ep

1.IT~1t

DH(X) < /30

4 + ~.

Then for any

R ~ Ro

+1

there

Ro was defined in Remark 4.7.

Proof:

Suppose by contradiction that for -R

R

= Ro + 1 411'

-

/3 := /3 = DH(X) < /30 + 3H2 • Let E = DH(X) - DH(X l ) > 0, and let N be a neighborhood of Jet' as in Lemma 4.6 and Remark 4.7. Choose E> 0 and a deformation cJ according to Lemma 4.8, and let PEP satisfy sup EH(:Z:) < f3 _Ep

1-ITs a

+ E.

133

IV. Unstable H-aurfaces.

By property i) of ~ the deformed path p'

= ~(p, 1) E P.

By iii), moreover,

Since N nMf:!-E = 0 by Lemma 4.6, while by Remark 4.7 N n {:z: E M 11:z:IT ~ Ro} = 0, we conclude that either p' eN or p' n N = 0. But z E p' n N while :Z:1 E p'\N. I.e. p' intersects lemma.

N but is not contained in N. The contradiction proves the

o

=

=

We now return to the case X XH. Note that by Theorem 111.2.1 for :z: (:Z:o, z) EM uniformly bounded in M also V(X(:z:)) remains uniformly bounded. In consequence, the functional EH is uniformly continuous in HEIR on any set {:z: E MI 1:z:IT :5 R}, and for H sufficiently close to our initially chosen H we have by Lemma 4.9:

f3-n:= inf suPEn(:z:) pEP zEp

(4.5)

> inf - pEP

sup

En(:c)

-Ep

> En(:CH),

l"IT$Ro+l

where P is defined by (4.4). Lemma 4.10: Proof:

-

The map H

Use the identity for

-+

0

':.ff f3n

is non -increasing.

< H1 < H2

and X E C(r) :

(4.6) Now suppose

H1 < H2 are sufficiently close to

Pm E P be a minimizing sequence for

sup EH1 (:c)

H1 -+

H such that (4.5) holds. Let

:

f3H1 (m

-+

00),

zEpm

and let :Z:m E Pm satisfy

EH2(:Cm) Applying (4.6) with Xm

= X(:Z:m}

= zEPm sup EH2(:C) ~ f3H2. we obtain that

134

B. Surfaces of prescribed constant mean curvature.

The lemma follows.

o By a classical result in Lebesgue measure theorey, Lemma 4.10 implies that the map

-H 1-+"1t ~ (4.7) 1£

is a.e. differentiable near H. Define

= {H E IR 1.8H

is defined near H and

lim sup fI ..... ll

!! < oo} . ( ~-'!:il) H - H

1£ is dense in a neighborhood of H. Therefore, we may approximate H by numbers Hm E 1£, Hm -+ H (m -+ 00). (If HE 1£, we may let Hm == H.)

Still maintaining our assumption (4.2) for our initially chosen H we now establish: Lemma 4.11: For any sufficiently large (fixed) satisfies the local Palais-Smale condition on M:

mE IN the functional EHm

=

Any sequence {X~hE.lV' X~ X(:z::;'),:z::;' E M, with D(X~) ::; c uniformly, DHm(X~) = EHm(:Z::;') -+ .8Hm , gHm(:Z::;') -+ 0 as (k -+ 00) is relatively compact.

Proof: By Lemma 4.5 the thesis is true unless for some sequence m -+ 00 EHm admits a critical point :Z:m with Xm = X(:Z:m) satisfying

By the construction of Lemma 3.4 we can estimate with a uniform constant

.8:

(4.8) for

m

2:: mo. Moreover, the a-priori bound Theorem 4.2 guarantees that D(Xm) ::;

for

m

3.8 + c(r) < 00

2:: mo. But then also

and by Lemma 4.5 {Xm} weakly accumulates at an H-surface X E C(r) with

DH(X)::; lim inf DHm(Xm) < .80' contradicting (4.2). m-+oo

135

IV. Unstable H-IIlU"fac:es.

o Lemma 4.12: For any sufficiently large m E lN there is a solution Xm. = X(zm) of the Plateau problem (111.1.1)-(111.1.3) for Hm, characterized by the condition DHm(Xm) =f3Hm' and Zm is a point of accumulation of a minimizing sequence of paths lN, such that sup EHm(z) -+ f3H m (1e -+ 00). "EP~

P!;.

E P, Ie E

Proof: Fix m E IN. Choose a sequence {H!hEN of numbers H! > Hm, H! -+ Hm (Ie -+ 00). Let {P~hEN' p~ E P be a minimizing sequence for Hm such that sup EHm(z) ~ f3Hm "EP~

(4.9)

For arbitrary

Z

+ (H! -

Hm).

E p~ with

(4.10) by (4.6)-applied to X

= X(z)-and (4.7) we obtain the uniform bound:

(4.11)

Suppose there exists 6> 0 such that for all (4.12) uniformly in

gHk

m

Z

E p~ satisfying (4.10) there holds

(z) ~ 6 > 0

Ie E IN.

By (4.11) and uniform continuity of E H , gH in H on bounded sets, for sufficiently large Ie a pseudo-gradient vector field for E H" near such Z will also be a m pseudo-gradient vector field for EHm near z, and a pseudo gradient line deformation of p~ near points satisfying (4.10) will yield a sequence of comparison paths still satisfying (4.9). So eventually (4.12) lets us arrive at a path pi E P where

136

B. Surfaces of prescribed constant mean curvature.

contradicting the definition of f3H". m Negating (4.12), by (4.9) - (4.11) we find a sequence {X!.

= X(z~)}

such that

D(X!.) ::; c , f3Hm ~ lim inf EHm(Z~) 1:-+00

lim gHm(Z~)

k-+oo

= lim

1:-+00

=

lim inf E H" (z~) ~ lim inf f3H k 1:-+00 m 1:-+00 m

= f3Hm ,

gH k (z~) -+ 0 (1: -+ (0), m

z~ EP~. By Lemma 4.11

{z~} accumulates at a critical point

Zm

of EHm.

D

Proof of Theorem 4.1: For Hm E 1£ tending to the solutions obtained in Lemma 4.12. By Theorem 4.2

H let Xm

= X(zm)

be

while by (4.8) we may assume that

and

By Lemma 4.5, assumption (4.2), and the definiton of f3o, the sequence {zm} is relatively compact and accumulates at a critical point z E M of EH. Moreover, z is an accumulation point of paths Pm E P where sup EHm(Z) -+ EH(Z)

( 4.13)

zEPm

=

If X X(Z) E C(r) were a relative minimum of DH, DOW (4.13) would give a contradiction to Lemma 4.9. Hence (4.2) cannot be true, and the proof is complete. (]

137

IV. Unstable H-surfaces.

Finally we present the proof of Theorem 111.3.4. Recall the assertion: Theorem 4.13: If r is a Jordan curve of class C 2 in JR3, HE JR, and if for some X E C(r) there holds 2

~

H D(X)

2

< 3?1',

then DH admits a relative minimum conditions that D(XH ) DH(XH)

XH

on

C(r) characterized by the

< 5D(X),

= min { DH(X) I X

E C(r), D(X)

< 5D(X) } .

Proof of Theorem 4.13: By Theorem 1.4.10 we may asssume that minimal surface. Moreover, it remains to consider the case H::j;O.

X

is a

Let

M = {:z: EM I D(X(:z:)) < 5D(X)}. Define

Claim 1:

f30>

-00.

Let :z: E M, X = X(:z:). Applying a variant of the isoperimetric inequality, (cp. Remark 1I1.2.2.ii), we may estimate 1

IV(X)I

~ IV(X)I + [ 36?1'

+ D(X»)

(D(X)

3] 1/2

~ C

< 00,

uniformly, and the claim follows. Claim 2:

f30 < inf {DH(X) E c(r), D(X) = 5D(X) } =: {3

Simply estimate, using the isoperimetric inequality DH(X) - DH(X) =D(X) - D(X)

~4D(X) - 2

( 4.14)

HII¥-

= SD(X).

V(X))

1

~4D(X) (1if D(X)

+ 2H(V(X) -

2:

0,

138 Since

B. Surfaces of prescribed constant mean curvature.

X

is a minimal surface, while HiO, it follows that

and there exists a surface X E X

+ HJ,2(Bj llf)

such that D(X)

Now remark that Lemma 4.5 implies that any sequence {:Z:m}

< 5D(X) and

eM

such that

EH(:Z:m) -+ Po, gH(:Z:m) -+ is relatively compact, i.e.

°

EH satisfies the Palais-Smale condition

(P.S. ){30 on

M. Indeed, by Lemma 4.5 we may assume that Xm = X(:Z:m) !£,. Xo = X(:z:). By weak lower semi - continuity of Dirichlet's integral X satisfies D(X) ~ 5D(X). In particular, :z: E M, EH(:z:) ~ Po, and by Lemma 4.5 Xm -+ X strongly as m -+ 00.

Finally, suppose by contradiction that (P,S'){3o there exists 60 > such that

°

Po

is a regular value of

EH on

M.

By

(4.15)

For 6

and let EH on

>

°

let

M6 = {:z: EM I EH(:Z:) < Po + 6},

e: M6 -+ T be a Lipschitz M6 satisfying the conditions

continuous pseudo-gradient vector field for

0

(4.16)

e(:Z:)+:Z:EM, le(:z:)lr < 1 (dEH(:z:),e(:z:»)