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 +E
VXV
B
Hence the first stationarity condition i) is equivalent to the condition
f B
VXV
= 0,
V
H~,2(B;lR")
9
I. Exiltence of a solution.
which in turn is just the weak form of the differential equation (1.1). By standard regularity results any weak solution X E Hl,2(BjJRn ) of (1.1) will be smooth in Band (1.1) will be satisfied in the classical sense. It remains to show that for harmonic X E C(r) the stationarity condition ii) is equivalent to the conformality relations (1.2). This result requires some preparatory lemmata which we state in a slightly more general way than will actually be needed. Lemma 2.3: Let G be a domain in JR2 =([;' and suppose X E Hl,2 (Gj 1K') is harmonic. Then the function
is a holomorphic function of w ~
Note that
Proof:
= u + iv E G C ([;' .
may be written as a product
with component-wise complex multiplication and
the usual complex differential operators. Note that
lia Hence by harmonicity of X a~
i.e.
~
= 2aaX·aX
2t:.X·
ax
0,
is holomorphic.
o Lemma 2.4: Suppose G is a domain in JR2, and let X E Hl,2( Gj JRn). Moreover, suppose that for any differentiable family of diffeomorhisms ge: G -+ G e with go = id there holds
Then X is conformal. Proof:
Let
T
E C 1 (Gj.lR 2 ) and for
e E.lR with
10
A. The claasical Plateau Problem for disc - type minimal surface..
consider maps g€ = id + €T: G --+ G€ := gdG). Since by choice of € the maps g€ are injective and the rank of the differential
is maximal everywhere the g€ in fact are diffeomorphisms
g€: G
--+
G€.
Compute by the chain rule:
D(X
0
gil; G€)lf=O
=~
IIV(X
0
gi l )12 dw
G€
=
~
II
((V X)
gil) . Vgi l 12 dw
0
Gf
=~
IIVX.((Vgi l )ogf)1 2
det(Vg€)dw
G
Now gil
0
g€ =
while - labeling
id implies that
T
= (Tl, r2)_ det(Vgd
I.e.
D(Xogi l ; Gf )
=1/2/IVXI2-2f(lXuI2r~
+IX.. 12r; + Xu·X.. (r:+r;))
G
+ fIVXI2(r~ + r;)dw + 0(f2). It is now clear that
f
f-+
D(X
0
gil; G f ) is differentiable at f = 0 and
d -1 d€ D(X 0 g€ ; Gf)bo = =
~
1(IXuI2-lx.. 12)(r~-r;)+2 Xu·X .. (r~+r;)dudv. G
If now again we consider IR 2 =.
where
~
w = u + iv, r = rl + ir2 we may
is defined as in Lemma 2.3.
Thus
(2.6)
:€D(X 0 gil; Gf)/€=o = -1/2
! Real(~. G
8r)du dv
11
I. Existence of a solution.
and the expression can only vanish for all 'T E C 1(G;JR 2) if ~ vanishes identically in G, i.e. if X is conformal.
o To conclude the proof of Lemma 2.2 in view of Lemma 2.4 it suffices to remark that by (2.6) conformality of X also implies the stationarity condition ii) of Lemma 2.2. Hence the critical points of D in C(r) precisely correspond to the solutions of Plateau's problem.
o Remarks 2.5. i) If X is harmonic on integrating by parts in (2.6) we obtain :f D(X o9i 1 ; Bd If=o
j
= -~
B
, by Lemma 2.3 and upon
Real (~. W'T) do.
DB
Thus, the conformality relations (1.2) may be interpreted as a natural boundary condition for the holomorphic function ~ associated with X. Cpo Courant [1, p. 72 tfJ.
ii) Variations of the type i) in Lemma 2.1 may be interpreted as "variations of the dependent variables" i.e. of the 6urface X. Variations of the type ii) ("variations of the independent variables") correspond to variations of the parametrization of X.
iii) By conformal invariance of D and the Riemann mapping theorem any minimizer Xo of D in C(r) will be a critical point of D in the sense of Lemma 2.1. Indeed, by (2.6) it suffices to show that Xo satisfies the stationarity condition ii) of Lemma 2.2 for all 9f = id + f'T, 'T E C 1(B; JR2). Suppose by contradiction that for some 'T E C 1(B; JR2)
with Bf have
= (id + f'T)(B).
Then for some f D(Xf; Bd
But
Bf is conformal to
<
f:.
0 and
Xf
= Xo 0 (id + f'T)-l
we
D(Xo).
B. Hence we may compose X f with a conformal map Xf = X f 09f E C(r) with
gE : B -+ Bf to obtain a comparison surface
D(Xd = D(Xf; Bd
<
D(Xo) = min{D(X)IX E C(r)}.
The contradiction proves that Xo is critical for
D.
12
A. The classical Plateau Problem (or disc - type minimal surfaces.
3. The direct methods in the calculus of variations. We now proceed to derive the existence of a minimizer of D on C(r)-and hence of a solution to Plateau's problem (1.1) - (1.3), cpo Remark 2.4. iii) - from the following general principle: Theorem 3.1: lRU{oo}.
Let
M be a topological Hausdorff space, and let
E: M -+
Suppose that for any a E lR the set
Ma := {:z: E M I E(:z:) :::; a}
(3.1) is compact.
Then there exists :Z:o E M such that E(:z:o)
= "'EM inf E(:z:).
In particular, E is bounded from below and lower semi-continuous on M. Proof:
Let ao
and consider a sequence m-+oo .
= "'EM inf E(:z:)
{am} of numbers
By compactness of Mal for
~ -00,
am
>
ao
tending to
ao
as
a E lR, the nested sequence
has non-empty intersection and there exists :Z:o E
n
M am •
mEN
Clearly, aO:::; E(:z:o) :::; am for any m and therefore letting that E(:z:o) = ao. Since E does not assume the value bounded from below on M.
-00,
in particular ao
Finally, by (3.1) for any a E lR the set {:z: E M lower semi-continuous.
I E(:z:) > a}
m -+
>
-00,
00
we infer
and
E
IS
is open, i.e. E is []
13
I. Existence of a solution.
Remark 3.2: In the work of M. Morse compactness of the sets M", in Theorem 3.1 defines the property of "bounded compactness" of E on M. This condition implies lower semi-continuity of E. However E cannot be continuous on M and simultaneously satisfy (3.1) unless M is locally compact: By (3.1) any set
M",
= {:t: EM I E(:t:) < a}
must be relatively compact in M while by continuity M", is also open. In applications a simple variant of Theorem 3.1 will often be encountered: Theorem 3.3: Suppose M is a sub-set of a separable Hilbert space H which is closed with respect to the weak topology on H.
Let E: M -+ IR be a funtional which is sequentially weakly lower semi-continuous on M, i.e. which satisfies the condition
(3.2)
E(:t:) < lim inf E(:t: m ), m-+
Also assume that there holds:
whenever:t:m!£':t:,:t: m E M.
00
E is coercive, i.e. suppose that for any sequence {:t: m} in M
Then there exists a minimizer :t:o E M with
E(:t:o)
= ",eM inf E(:t:).
Theorem 3.3 is reduced to Theorem 3.1 by letting M be endowed with the weak topology on H. However, there also is a very natural direct and constructive proof of Theorem 3.3 which uses the concept of a minimizing sequence. Proof of Theorem 3.3:
Let ao
= zeM inf E(:t:)
;:::
-00,
and let {:em} C M be a sequence such that E(:t: m ) -+ ao as m -+ 00. By coerciveness of E {:t: m} is bounded and hence weakly relatively compact. Extracting a weakly convergent subsequence :t: m !£. :t:o, by weak closedness of M also the weak limit :t:o E M. Finally, by (3.2)
and the proof is complete. []
14
A. The classical Plateau Problem for disc - type minimal surfaces.
Examples 3.4:
i) the norm in a Hilbert space
H with scalar product
(-,.)
is weakly lower semi-continuous. ii) More generally, let a: H x H on H such that
-+
m
be a continuous symmetric bilinear form
a(z,z):2: 0, Then
VzEH.
E(z) == a(z, z) is weakly lower semi-continuous on H. H 1 ,2(Bj mn).
In particular, D is weakly lower semi-continuous on
Proof:
Suppose
Zm
~
z. Then
By the Riesz representation theorem there exists a(z,zm - z)
= (y,
y E H such that
Zm - z)
-+
o.
o iii)Suppose E:H-+mu{oo} iscontinuousandconvez,Le. for all z,yEH 0< a < 1 there holds
E(az
(3.4)
+ (1- a)y)
:::; aE(z)
+
(1 - a)E(y).
Then E is weakly lower semi-continuous on H.
Proof:
If Zm
W
1 zN:= N
z weakly, by the Banach-Saks theorem
N
L
Zm
-+
z
strongly in H
as N
-+ 00.
m=l
Hence, by continuity of E and (3.4)
E(z)
lim inf E(zm). m ..... oo
15
I. Existence of a solution.
Remark:
The inequality 1 ::; N
N
L: E(zm),
{Zm} C H, N E IN
m=l
for a convex functional
Example 3.5:
E: H
-+
The functional
IR U {oo} is a special case of Jensen's inequality.
D is coercive on C(r).
Proof: By the Sobolev inequality for L OO ( aE; IRn) :
X E H 1 ,2(B;IRn )
with
IIXII~ ::; c (IIV XII~ + IIXII~,OB) ::; c (D(X) + IIXII~oo(OB») . Hence for
X E C(r):
IIXIIi::; c D(X) + c(r).
XIOB E
16
A. The classical Plateau Problem for disc - type minimal surfaces.
4. The Courant Lebesgue Lemma and its consequences. In the preceding chapter we have seen the importance of weak closedness of C(r). However, the presence of the conformal group of the disc
G = {g(w) =
(4.1)
ei 4>o
I
a+_w a E l+aw
0 E IR}
acting on C(r) and conformal invariance of D cause problems.
=
Lemma 4.1: For X E C(r) let X 0 G {X 0 gig E G} be the conformal orbit of X. Then for any X the weak closure of X 0 G contains a constant map. Proof:
am E
i) First consider
Clearly, as m -+
-+
t.p E C1(Bj IRn). Let
gm () w
h = 1am+w + amw ,were
1.
00
gm(W) for all wEB, uniformly away from
w
-+
1
= -1.
Hence pointwise in
B.
By conformal invariance of D moreover
while It.pm IL oo
=
1t.pIL oo <
00,
and {t.pm} admits also a weakly convergent subsequence proves our claim for regular functions. ii) For X E C(r) let
t.pm !£.
t.po
== t.p(I).
This
gm be as above and define Xm
=X
0
g;;,,1 E C(r).
By (2.2) and Example 3.5 {Xm} is bounded in H 1,2(BjIRn ) and we may extract a subsequence Xm !£. Xo. To show that Xo
==
const it suffices to show that
J
VXoVt.p.dw
= 0,
'V t.pEC1(BjIRn ).
B
But by i) of this proof and conformal invariance of D, with have: = lim jVXmVt.pdW m-+oo
B
= m-+oo lim jVXVt.pm dw = 0, B
t.pm
=
t.po gm we
17
I. Existence of a solution.
which completes the proof.
o In view of Lemma 4.1 the set C(f) cannot be weakly closed in H 1 ,2(B;JR"). However, equivariance of D with respect to G allows us to factor out the symmetry group. The most convenient way to do this is by imposing a three - point _ condition on admissible functions. Note that (4.1) immediately implies: Lemma 4.2: < 211", 0 ~ 1/J1
1,03
Given any triples (1/>1,1/>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
<
it/>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
<6$
60 , such that
Iln612':
1611"~(X) fl
Let
P E [6, V6] , C p
Cp(wo) be selected according to Lemma 4.4 satisfying
J
cp
IX.12ds $ 8D(X) . pllnpl
=
Denote Wi = e21ri 4>i, j = 1,2, the points of intersection of Cp with fJB, Cp fJB n Bp(wo ) that subarc of fJB with end-points Wl> W2 which contains at most one of the points Pi' j 1,2,3. Also let Xi X(wi), j 1,2, and let f be that subarc of r connecting Xl> 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», rm = 'Ym(BB). In this .f1.
way we generate a sequence of C 2 -diffeomorphisms such that
Extending the parametrizations 'Ym to harmonic surfaces Xm we immediately see that C(rm);i:0 for all m. By Proposition 4.6 there exist solutions Xm E C(rnJ to ~1.1) - ( 1.3) for r m, moreover, by Theorem 5.1 below Xm E C 1 (Bj IRn + ). But then Theorem 4.9 assures the bound (4.4) for large m. Now let Qj = 'Y(Pj ), Q;m) = 'Ym(Pj ), j = 1,2,3, mE IN. With no loss of generality we may assume that Xm E c*(r m) where we normalize with reference to the triples (Q;m»,j = 1,2,3. Since for this normalization the constants m ) appearing in the proof of Lemma 4.3 clearly have a uniform lower bound El > 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)
0
~
I,
be a branch line of X along which X(-Yl(t)) = X(-y2(t)) while the u-derivatives of X along 'Yl, 'Y2 are transverse. The figure illustrates the two - stage process of transforming X into a surface Y with the same area as X by a discontinuous transformation of the parameter space. (The successive images of 'Yl, 'Y2 are indicated for clarity.)
-II
11
/
\
12
12
11
•
\
/
•
12
After this change of parametrization has been carried out the reparametrized surface Y will still belong to the class C(r). Instead of a branch line, however, where two sheets intersect the new surface Y will have a contact line where two sheets touch. Moreover each sheet contains an edge along the contact line. Area can hence be reduced by smoothing off the edges. In this way we obtain a comparison surface Z E C(r) such that
A(Z) < A(Y)
= A(X) = D(X).
On the other hand, by (2.5) and since X by assumption minimizes D
D(X)
in
C(r)
= X' inf D(X') = inf A(X'). ec(I') X' ec(I')
A contradiction. For branch points on analytic boundaries the same reasoning applies, cf. GulliverLesley [1]. The essential ingredient again is the existence of a branch line through the branch point. For 6mooth wires R. Gulliver [2] has recently presented an example of a smooth wire spanning a minimal surface with a boundary branch point which is not connected to a branch line of the surface.
I. Existence of a lolution.
27
Finally, let us mention a result of Sasaki [1] and Nitsche [1, §380 , formula (156)] which relates the number and order of the branch points of a minimal surface X E to the total curvature of its boundary r by the Gauss-Bonnet formula and hence permits to estimate the former independently of X.
nt
Theorem 5.9: Let r be a Jordan curve in IIf of class C 2 with total curvature /C(r). Suppose X E C(r) solves (1.1) -(1.3) and has interior branch points wi, of orders Ai, 1:::; j :::; p, and boundary branch points ei 4>1: 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
~ 'lDoEB sup { sup Ig(w) 'ID,w'EBp(wo)nB
~ 'lDoEB su~ { 'ID,'Is~p Ig(w) D ECp
g(W')I}.
g(W')I} < 1-'.
This proves equi-continuity of H1,2-bounded subsets of :F. In particular, any g E:F is a continuous map of B onto B oB monotonically onto oB and satisfying the three-point-condition g 2~ik
ma~ping
(e¥)
=
--
e-Y-, k=l,2,3. Thus,for ZEC·(r)nC 2 (B;lRn) also ZogEC·(r). Moreover, if gm. !£. g weakly in H1,2(B; 1R2), gm. E:F, we have
(A.2)
det(Vgm) = g;'u g;'" - g;'"g!.u = = (g;'g!.,,)u - (g;'g!.JlI !£. (glg~)u - (glg;)"
in the sense of distributions. We use this fact to compute gm. - g uniformly there holds
A(Z)
= A(Z
0
gm)
j
= Vdet «V Z· V Z)
= det(Vg)
A(Z 0 g). First, since
0
gm) det(Vg m ) dw
0
g) det(Vg m ) dw + 0(1)
B
=
j V det «VZ. V Z)
B
with 0(1) - 0 . By (A.2) this equals
= -
j
(g;'g!.l1 Ou - g;'g!.u 0,,)
Vdet «V Z . V Z)
0
g) dw
B
+
j g;,oTg!.Vdet«VZ.VZ)og)do+o(l),
BB
OT denoting the tangent derivative in counter clock-wise direction. Since gm. maps oB monotonically onto oB we have OTgm. E L1 with J IOTgmido = 21f. Hence we may pass to the limit m -
BB
00
A(Z 09)
in both integrals and finally arrive at the identity
= mlim A(Z .... oo
0
9m)
= A(Z),
31
I. Existence of a solution.
as claimed.
o Proof of Proposition A.l (completed): Now consider the functional E on :F: E(g) = D(Xf
0
g) =
~
J
I(VXf )
0
g). Vgl 2 dw.
B
By choice of X f we have E(g) ~ fD(g)
and by Example 3.5 E is coercive on :F with respect to the Hl,2 -topology. Let gm E:F be a minimizing sequence such that E(gm)
-+
in£E(g). gEF
By Proposition 4.5 we may assume that
gm ~ 9 E:F and X f
0
g...,. ~ X f
0
9 E
c·(r), whence by weak lower semi-continuity of Dirichlet's integral D(Xf
0
g)
= E(g) ~
lim inf D(Xf
0
gm)
=
m-+ 00
lim inf E(gm). m-+ 00
I.e. Xf 0 9 nummlzes D among all surfaces Xf 0 g', g' E :F. In particular, X f 0 9 satisfies the hypothesis of Lemma 2.4 and is conformal. But then by Lemma A.2 A(Xf) = A(Xf 0 g) = D(Xf 0 g) = inf D(Xf 0 g'), g' E"J!
proving (A.l) and the Proposition.
D
II. Unstable minimal surfaces 1. Ljusternik-Schnirelman theory on convex sets in Banach spaces. The method of gradient line deformations and the minimax-principle are the most general avaible tools for obtaining unstable critical points in the calculus of variations. Historically, the use of these methods can be traced back to the beginning of this century, cf. Birkhoff's [1] theorem on the existence of closed geodesics on surfaces of genus 0 . Through their famous improvement of Birkhoff's result the names of Ljusternik and Schnirelman [1] became intimately attached to these methods. In 1964 a major extension of these techniques was proposed by Palais [1], [2], Smale [1] and Palais - Smale [1]. Their fundamental work has found many applications and has inspired a lot of further research. One of the most significant contributions may be the well-known paper of A. Ambrosetti and P. H. Rabinowitz [1]. For the Plateau Problem, however, these methods still seemed inadequate, and for a long time analysts believed that the Palais-Smale condition would "never" be satisfied in any "hard" variational problem as the Plateau problem for minimal surfaces was considered to be. (Cf. the remarks by Hildebrandt [4, p. 323 f.J) As we shall see, by a simple variation of the classical concepts of Ljusternik-Schnirelman, resp. Palais and Smale the Plateau problem can be naturally incorporated in the frame of these methods. This extension of Ljusternik-Schnirelmann theory and its application to the Plateau problem was presented in Struwe [1]. In abstract terms we may regard this method as an extension of Ljusternik-Schnirelman theory to functionals defined on closed convex sets of (affine) Banach spaces and satisfying a variant of the Palais - Smale condition.
We now recall the pertinent ideas. Throughout this section we make the following
Assumption: Let T be a Banach space with norm I· I, MeT closed and convex. Suppose E : M --+ IR admits a Frechet differentiable extension E E C1(T; IR).
Definition 1.1:
At a point z E M let
g(z)
=
sup yEM
Iz - yl < 1 measure the ,lope of E in M.
(dE(z), z - y}
34
A. The classical Plateau Problem for disc - type minimal surfaces.
Remark 1.2.
i) If M
= T,
then g(z)
= IdE(z)l.
ii) Since E E C 1 and M is convex, 9 is continuous.
Definition 1.3: A point z E M is called critical iff g(z) = O. Otherwise z is called regular. If z is critical, (3 = E(:c) is called a critical value of E. If E- 1 ({3) consists only of regular points, {3 is called regular. Remark 1.4: If M = T by Remark 1.2 Definition 1.3 coincides with the usual definition of critical points.
Definition 1.5: Let M = {:c E Mlg(z)iO}. A Lipschitz continuous mapping -+ T is a pseudo gradient vector field for E on M if the following holds:
e:M i) ii)
e(:C)+:CEM,
V ZEM,
There exists a constant c > 0 such that a) le(:c)1 < 1, b) (dE(:c), e(z») < -min{c- 1 g(:c)2, I}.
Exactly as in the case M
Lemma 1.6: Proof:
Let
= T,
cf. Palais [3, Chapter 3], we now establish
There exists a pseudo-gradient vector field Zo E
M
e for
E on
M.
and choose Yo E M such that
< min{g(:co), I}, (dE(zo), :Co - Yo) > min{I/2 g(:CO)2, I}. l:co - yol
(1.1)
By continuity, there exists a neighborhood V(:co) of :Co in M such that for :c E V(:co) we still have l:c - yol < min{g(:c), I} while (dE(:c),:c - Yo) > min{I/2 g(:c)2, I}. Hence eo(:c):= Yo -:c is a pseudo-gradient vector field for E on V(zo). The sets {V(:CO)"'OEM are an open cover of M. Since MeT is metric there exists a locally finite refinement {V(:C')}'EI of this cover, i.e. having the property that for any :Co E M there is a neighborhood YO of :Co and a finite collection 10 C 1 such that for L E 1\10 we have v;, n V(:c.) = 0, cf. Kelly [I, Thm. 8, p.156j Cor. 35, p. 160]. Let {1/>'}'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
~
1 such that
and let 1/ be a Lipschitz continuous function on M such that 0 ~ 1/ ~ 1 and == 0 in Nf3,6, 1/ == 1 outside N f3 ,6'. Also let f < 1/2 min{o,E} be a number to be specified later, and choose a function cp E C(f(lR) such that o ~ cp ~ 1, cp(s) = 1 if Is -.61 < f, cp(s) = 0 if Is-.6I;::: 2f.
1/
Define
e(z)
= { cp(E(z»
;(z)e(z),
zEM , else,
where e is the pseudo-gradient vector field constructed in Lemma 1.8. Since any critical point of E on M has a neighborhood where either cp or 1/ vanishes, e is Lipschitz continuous. Moreover, by convexity of M, e satisfies
e(z) + z E M
(1.2) at any point z E M.
Now let
(1.3)
~:
[0, oo[x M
-+
M be the solution to the initial value problem
a
at ~(t, z) ~(O,
z)
= e(~(t, z», = z.
By convexity of M, i) may be constructed as a limit of approximate trajectories of (1.3) by Euler's method, cpo Struwe [1, Lemma 3.8].
II. Unstable minimal swfaces
37
If T is locally strictly convex and if the projection Pm: T M obtained by letting IPM(:Z:) -:z:1 = inf Iy -:z:1
-+
M of
Tonto
yEM
is locally Lipschitz continuous (e.g. if T is a Hilbert space), then a more instructive existence proof goes as follows: Extend e to T by letting
= e(PM(:Z:» .
e(:z:)
Now let "¥: [0, oo[x T -+ T be the solution to (1.3) on T which exists globally by Lipschitz continuity and bounded ness of e. By (1.2) M is an invariant region for "¥ and the deformation ~ may be obtained by restricting "¥ to M. ~ solves (1.3), each ~(t,.) now is a continuous map from Minto M and trivially satisfies i), ii) by our choice of £,1/. Finally, if E(:z:) S 13 + £, either E(~(I,:z:» S 13 - £ or IE(~(t,:z:» - 131 S £ for all t E [0,1]. In the latter case, moreover, by choice of 1/:
As ~
J S 13 + + J1/(~(:Z:, 13 + J 1
E(~(I,:z:» = E(:z:) + :tE(~(t,:z:»dt o
1
(1.4)
£
t»(dE(.), e(.)}dt
o
1
S
£ -
S 13 + £ Now suppose :z: (/. N or all t E [0,1], or the flow U{3,p\U{3,P/2. By boundedness
-
1/(-) min{I/2 g( .)2, l}dt o 1/2 h2 . I{t E [0, 111~(t,:z:) (/. N{3,6'}1.
~(I,:z:)
(/. N. Then either ~(t,:z:) (/. N{3,6' for through :z: must traverse the annulus le(:z:) I S 1 this will require ~(t,:z:)
I{t E [0, 1]1
~(t,:z:)
(/. N{3,6'}1
~
p/2,
and hence in any event we obtain that
Thus iii) will be satisfied if we let
This completes the construction. []
A variant of the deformation lemma 1.9 yields the following result:
38
A. The cla.saical Plateau Problem for elise - type minimal surfaces.
Lemma 1.11: Suppose E satisfies (P.S.) on M, and let :Z:o E M be a strict relative minimum of E in M. Then there exists a number EO > 0 such that for any E E]O, EO[ we have inf E(:z:) > E(:z:o). mE'"
I:z:- :z:ol =E Proof:
By assumption there exists EO
E(:z:) > 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, <1.> be defined as in Lemma 1.9. Consider the sequence Ym:= <1.>( :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(<1.>(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 <1.>(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(
h is linear and continuous by the maximum principle. coordinates on BB. Then the map
(2.1')
Also let
ei.p
denote
X:zl-+h(-yoz),
where 'Y 0 z denotes the map
(2.1") is a local homeomorphism between
(2.2)
M
= {z E CO(IR;
IR) 1 z monotone; z(¢ + 27r) = z(¢)
and Co(r) with respect to the in a similar way the set C~(r)
+ 27r,
V ¢; D(X(z»
< co}.
CO-topology. Introducing a three point-condition,
= {X E C*(r)16X = O}
can be represented via (2.1) in terms of
2d V kE~}. M * ={zEMlz (2d ""3 ) =-3-'
(2.3)
Finally, for z E M let
JJ 2,.. 2,..
(2.4)
E(z)
= D(X(z» =
_1
167r
°°
,
2
h(z(¢» - 'Y(z~¢ »1 d¢ d¢', . SIn
2
(¢ -¢) -2-
denote the Douglas - Dirichlet integral of X(z) = h('Y 0 z), cf. Nitsche [1;§§ 310 311].
42
A. The classical Plateau Problem for disc - type minimal .urfaces.
Proposition 1.4.7 in this notation takes the form: Proposition 2.1:
For any a E IR the set
{Z E M·
I E(z) ~ a}
is compact with respect to the CO-topology.
For the larger set M we have the following weaker compactness result which we note for later convenience: Propsition 2.2: Suppose {zm} is a sequence in M such that IZm(O)1 ~ c uniformly. Suppose E(zm) ~ c uniformly. Then a subsequence Zm -+ Z in LII (IR) and either oc z(¢I)== const (mod 211"), or
Zm and
Z
-+
Z
uniformly (m
-+
00),
E M.
Proof: Local Ll-compactness of {zm} follows from Helly's theorem and monotonicity of Zm. Note that Z is monotone; hence if Z is continuous then Zm -+ Z uniformly by Dini's theorem, and Z E M. In any event, letting Xm = X(zm), X = X(z) = h(,),oz), by uniform boundedness of E(zm) and weak lower semi-continuity of D: E(z) < 00. But finiteness of Douglas' integral (2.4) and monotonicity of Z imply continuity of ')' 0 z. Hence, if Z is discontinuous it follows that ')' 0 Z == const ,i.e. Z == const. (mod 211").
o Note that by weak lower semi-continuity of D on H 1 ,2(B) the functional E will be lower semi-continuous on M, if we endow M with the topology of co. We now leave the trails of Douglas and his followers and define a new topology on M which will render E differentiable (and even smooth for smooth r)! Throughout the following assume ')' is a diffeomorphism of class C r Note that
,
r ~ 2.
M is a convex subset of the affine space
However, M is not closed in the induced topology (if we only allow finite values of E ).
43
II. Unstable minimal surfaces
Now let
T
(2.5)
= H 1/ 2 ,2 n cO(IR/21r),
where the seminorm in H 1/ 2 ,2(IR/21r) is given by 4,.. 4,..
( 2.6)
1:e12 1/2
= 111:e(tjJ)ItjJ -_
o 0 and the norm 1·1 in T is that induced by
I· 11/2 and
I· IL oo •
{id} + T.
M is a closed convex subset of the affine space
Lemma 2.3:
E(:e) < 00 implies that the expression
Proof: It suffices to show that finiteness (2.6) is finite and vice verse. But if [s]
:e(tjJ')12 dA. dA.1 tjJ'12 't' 't'
= inf{ls -
2m1rllm E~} the integral (2.4) 2,.. 2,.. I 2 E(:e) _1 I I h'(:e(tjJ)) -1'(:e~tjJ ))1 dtjJ dtjJ'
=
161r o
.2
Sln
0
(tjJ-tjJ) -2-
is equivalent to the expression 2,.. 2,..
1I o
I1'(:e(tjJ)) _1'(:e(tjJ')W . [:e(tjJ) - :e(tjJ'W dtjJ dtjJ'. [:e( tjJ) - :e( tjJ')] 2 [tjJ - tjJ']2
0
The latter in turn may be estimated from above 2,.. 2,..
< 11~'V112 .11 [:e(tjJ) - :e(tjJ')]2 dA. dA.1 -
dtjJ'
[tjJ _ tjJ']2
00
o
< II~ -
't'
't'
0
4,.. 4,..
I
2
112 .111:e(tjJ) - :e(tjJ )1 dA. dA.l. dtjJ1'oo ItjJ-tjJ'12 't' 't'
o
0
To obtain an estimate from below note that since exists a constant c., > 0 such that
11'(~:=~~tW ~ c."
V s,t
0
is a diffeomorphism there
l'
< Is-tl < 1r.
Hence
~c.,
4,.. 4,..
JJ(::) o
0
2
dtjJ dtjJ',
44
A. The classical Plateau Problem for disc - type minimal surfaces.
where (2.7)
¢'"
= max{~ > Ollz(¢ +~) -
z(¢)1 <
7f,
is a constant depending on (the modulus of continuity of) Therefore, for monotone z we have E(z)
< 00 iff z
V¢} > 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
0
which is equivalent to the usual definition
IXI BB 121/2
(2.8)
-!! -
IX(w) - X(w')1 2 dw dw' Iw-w'1 2 '
8B8B
cr.
Adams [1, Theorems 7.48, 7.53].
It remains to show that the map z 1-+ 'YOZ given by (2.1") extends to a differentiable map of class C .. - 1 of {id} + T into H 1 / 2 ,2 n LOO(BBjJR") if l' is a diffeomorphism of class C", r ~ 2. But this directly follows from the chain rule, the pointwise representation
d'
dz' (1'
0
z)(6, ... ,~,)
and the following elementary
Lemma 2.6: i) For any eT E C 1 (JRm j JRn ), P E H 1 / 2 ,2 n L""(BBjJRm composition eT 0 P E Hl/2,2 n L""(BB : JRn) and
)
the
leT 0 ph/2 :::; II(VeT) 0 plloo . Iph/2' ii)Forany ~,1/EH1/2,2nL""(BB) the product ~'1/EH1/2,2nL""(BB) and
iii) For any ~ E H 1/ 2 ,2(BB), 1/ E C 1(BB) the product ~'1/ E H 1/ 2 ,2(BB) and I~ '1/11/2 :::;
:rhecomplete. proofofLemma
d
lIelio IId
1~ll/2 •
2.6 is immediate from (2.8). Hence also the proof of Lemma 2.5
18
I]
In particular, from Lemma 2.5 we obtain
.Lemma 2.7:
The functional E: 'M C"'-1 on {id} + T, if l' E C", r ~ 2.
-+
IR extends to a functional of class
46 Proof:
A. The classical Plateau Problem for disc - type minimal surfaces.
E(z)
= D(X(z)),
D is a quadratic (analytic) functional.
and
o In order to apply the abstract tools developed in the preceding section we will need a compactness property like Proposition 2.1 . However, for reasons that will become clear in Section 11.4, we prefer to introduce a different normalization with respect to the conformal group action than the three-point-condition introduced earlier. Let G be our representation (1.4.1) of the conformal group of the disc with tangent space TidG span{iw, 1- w 2 , i(1 + w 2 )}.
=
In polar coordinates w
= rei-/>
an element 9 EGis represented by a map
9
and similarly TidG is represented TidG
= span{l,sin.p,cos.p}.
In the following, for ease of notation we will no longer distinguish between a conformal map 9 E G and the associated mapping g. Define
211"
Tt =
{e E T I / e· TJ d.p = 0,
VTJ E 11dG}
o and let
endowed with the topology inherited from T. Analogous to Proposition 2.1 we have:
Lemma 2.8:
For any
0:
E IR the set
{z E Mt I E(z) So:} is compact with respect to the CO-topology.
Proof: Let {zm} be a sequence in Mt such that E(zm) So: for some E IR and uniformly in m. By monotonicity and periodicity we have
0:
J 211"
Zm(O) S
2~
Zm d.p
o
= 11" S zm(211") = Zm(O) + 211" •
II. Unstable minimal.urfaces IZm(O)1 ~ 11"
Hence
47
,uniformly, and Proposition 2.2 implies that
Zm --+ Z
lD
Ltoc(IR) and either Z(¢) == canst (mod 211") or
E M and
Z
uniformly as m
Zm --+ Z
--+ 00 •
Moreover, in the second event, by lower semi-continuity of E with respect to uniform convergence in M, E(z) ~ a ,and the proof is complete in this case. To rule out the first possibility we introduce em normalization conditions
= Zm -
id, e
=Z -
id and use the
2,..
1
em . TJ d¢
= 0,
VTJ E T;d G .
o Since Zm --+ Z in L1 ,also em --+ e in L1 ,and also constraints which we may write down explicitly as follows: 2,..
2,..
2,..
1 =1 e d¢
e satisfies these
e sin ¢ d¢
=
1
e cos ¢ d¢
=0 .
0 0 0
But if
== canst
Z
(mod 211") ,and is 211"-periodic and monotone,
e is of the form
= c - ¢, 0 ~ ¢ ~ 211" . c = 11" ,while the second and third condition give
e(¢ - ¢o) The first condition now implies
1 =1 2,..
e(¢) (sin ¢ cos ¢o + cos ¢ sin ¢o) d¢
0=
o
2,..
2,..
e(¢) sin(¢ + ¢o) d¢
0
=
2,..
1
e(¢ - ¢o) sin(¢) d¢
0
The contradiction shows that the case proof is complete.
Z
=
-I
¢ sin¢ d¢ > 0 .
0
==
canst (mod 211") does not occur. The
o For
Z
E Mt let
g(Z)
(dE(z),
sup yE Mt
Iz - yl <
Z -
y)
1
as in Definition 1.1. Our approach is meaningful since we can identify critical points of E on Mt with minimal surfaces spanning r by means of the following
48
A. The classical Plateau Problem for disc - type minimal surfaces.
Proposition 2.9: A point z E Mt is critical for surface X = X(z) is a minimal surface spanning r. Proof:
E, i.e.
g(z)
= 0,
iff the
Compute
(dE(z), {)
d = deD(X(z+e{))le=o
! =! =! =
(2.9)
VX(z)V(:eX(X
+ e{)le=o)dw
B
8nX
:/r
0
(z + e{))le=odo
BB
8nX·
d~'Y(X).{ do,
V x E M, {E T.
BB solves (1.1) - (1.3), by Theorem 5.1 and since 'Y E C r , r ~ 2, X E Cl(B; lR''). Thus 8n X· ~'Y(z) == 0 on 8B by (1.2) and dE(z) = o.
If
X
To prove the converse implication for simplicity let us for the moment assume the following regularity result: Proposition 2.10:
If X E Mt solves
g(z)
= 0,
and
'Y E
cr,
r ~ 2, then
X = X(z) E H 2,2(B;lR"). Proposition 2.10 along with other regularity results will be established in Section 5 ofthis chapter.
Proof of Proposition 2.9 (continued): If X E H 2,2(B;lR") theorem, cpo Adams [1; Thm. 7.53,7.57], we have (2.10')
d 8"XIBB, d¢>XIBB
d d = «d¢>'Y)oz)·d¢>z E H l / 2 ,2(8B;
lRn) ~LP(BB;lR"), Vp<
But 'Y is a diffeomorphism. Hence also (2.10")
d~ z
For {E C l (lR/27r) such that
E LP(BB; lR"), Vp Ild~{lIoo
induces a diffeomorphism of BB
~
by the trace
< 00.
< 1 the map
lR/27r .
Fix {E C l (lR/27r) ,and for sufficiently small Cl(lR/27r) consider the parametrizations z(e,17)=zo(id+e{+17) .
€
and
00.
49
II. Unstable minimal surfaces
The map
F: 1R x TtdG ~ IIf given by
being continuously differentiable with ~~ (0,0) : Ttd G ~ 1R3 surjective, by the implicit function theorem there exists a differentiable mapping E 1-+ 17( E) in a neighborhood of E = 0, 17(0) = 0 such that
{or all E. Now, dE(:x) by (2.10) continuously extends to a functional on L2(oB) . Hence, differentiating by the chain rule and using the conformal invariance of E we obtain
J
= on X . ~'Y(:x) . (~:x. e) do d¢ d¢ 8B
=(dE(:x),
dd :XE) E
Note that we have used the fact that we obtain that
(2.11)
Jon
= - E_O+ lim .!.(dE(:x),:x E
E
,
e = J8nX . d¢d X . edo = 0
d 'Y(:X) . d¢:X d . do X . d¢
eE C (IRj27r) 1
2: o.
g(:x) = O. Hence, reversing the sign of
8B
for all
:x E)
8B
,and by density of such
e in
L 2 (8B):
onoB.
Now recall that by Lemma 1.2.3 the function
is a holomorphic function of w = u + iv = rei~ on B. Its imaginary part vanishing on aB, necessarily i is real on B and hence constant by the Cauchy - Riemann equations. Inspection at r = 0 now yields that
50
A. The classical Plateau Problem for disc - type minimal surfaces.
whence also ~(w) = (8X)2 = IXul2 -IX.,1 2 - 2iXu ' X.,
and
== 0
X is conformal.
o E on Mt.
Let us verify the Palais-Smale condition (P.S.) for
Lemma 2.11: Any sequence {:em} in Mt such that while g(:e m ) --+ 0 as m --+ 00 is relatively compact.
E(:e m )::::; c uniformly
Proof: By Lemma 2.8 and Remark 2.4 a subsequence :em --+ :e E M uniformly, X(:e m), X while {:em} is equi-bounded in H1/2,2. Moreover, letting Xm
=
~
X(:e), we may assume that also Xm On
X weakly in
H1,2(BjJRn ).
8B ,expand Xm - X = 'Y 0 :em - 'Y
:e
0
z
O"(y,z)
=
z
JJ
dd;2'Y(:e fl )d:e fl d:e 1
y ",'
By Lemma 2.6.i) the composed map
0" 0
Pm E H1/2
l1m11/2 = 10" 0 Pm11/2 ::::; II(VO")
0
,2
n L OO and
Pm 1100 ·IPmI1/2
::::; CII:em - :e1l 00 (I:emI1/2 Hence with error terms o( 1)
J
IV(Xm - XWdw
=
B
0 (m
--+
J J
--+
+ 1:e11/2)'
00)
8,.Xm(Xm - X)do
+ 0(1)
DB
=
8,.Xm ·
d~'Y(:em)(:em -
:e)do+o(1)
DB
= (dE(:em ), --+
:em - :e) + 0(1) :::; g(:em)I:e... - :el + 0(1) 0 (m --+ 00).
=
51
II. Unstable minimal surfaces
Thus Xm in Mt as
-+
X in H l ,2(B;lRn ) and Lemma 2.5 implies that m -+ 00.
Zm -+ Z
strongly
[]
Theorems 1.12, 1.13 now immediately imply the main result of this section, the "mountain-pass-lemma for minimal surfaces" due to Morse-Tompkins [1] and Shiffman [1] . Actually, the result below is a slight improvement of the Morse-Shiffman-Tompkins results in as much as oUI Theorem 1.13 also enables us to handle the case of relative minima which are not necessarily strict.
Theorem 2.12: Suppose a Jordan curve r of class C 2 bounds two distinct relative minima Xl, X 2 E C(r) . Then either D(Xt} D(X2) f30 and Xl. X 2 can be connected in any neighborhood of the set of minimal surfaces X spanning r with D(X) = f30 ,or there exists a minimal surface X3 spanning r distinct from Xl. X 2 which is not a relative minimum for D on C(r), i.e. which is an unstable solution of (1.1) - (1.3).
=
=
Remark 2.13: In particular, if r E C 2 bounds two distinct relative minima Xl.X2 E C*(r) at least one of which is strict, then r also bounds an unstable minimal surface.
=
Proof of Theorem 2.12: By conformal invariance we may assume that Xi X(Zi) with Zi E Mt, i = 1,2. Then Zl, Z2 are relative minima of E ,and by Theorem 1.13 and Lemma 2.11 either E(zt} E(X2) f30 and Xl, Z2 are connected in any neighborhood of Kf30 in Mt (hence in M), or there exists a critical point Z3 which is not a relative minimum of E in Mt (hence not in M ,either). Composing with X: M -+ C(r) we obtain the Theorem.
=
=
[]
52
A. The classical Plateau Problem for disc - type minimal surfaces.
3. Morse theory on convex sets. More detailed information about the number and types (relative minimum, saddle-point) of critical points of functionals can be obtained from Morse theory. In the 20's Marston Morse [1] developed a method for relating properties of the set of critical points of a functional with global topological properties of the space over which the functional is defined. Morse' ideas were recast by Milnor [1] in a more applicable form. A major extension of Milnor's approach to Morse theory was then made by Palais[1],[2], Palais-Smale [1] and Smale [1], in the 60's. Today a theory is emerging inspired by the deep work of Charles Conley [1] which seeks to combine the generality of Morse' original theory with the clarity of Milnor's approach. However, inspite of encouraging results by Rybakowski [1], Conley - Zehnder [1], and Rybakowski - Zehnder [1], Conley's version of Morse theory has not yet been conveyed to infinite-dimensional spaces. With applications to minimal surfaces in mind we proceed to develop a Morse theory for functionals defined on convex sets. Our presentation-based on Struwe [1] -to a large extent uses ideas and notions introduced by Milnor [1] and Palais [1]. However, in order to avoid coordinate transformations (which might destroy the convexity of M) normal form representations (the "Morse Lemma") are replaced by the use of pseudo - gradient vector fields, cf. Definition 1.7. Besides reducing regularity requirements on E this approch also seems more natural and simpler conceptually.
Throughout this section we make the following Assumption:
Let
M be a closed convex subset of an (affine) Banach space T with norm which is densely and continuously embedded in a Hilbert space H with inner product (-,.) and induced norm I· IH
I· IT
closed
(3.1)
MeT
dense <-+
H.
convex We moreover assume that M is also closed in H and that the topologies induced on M by H and T coincide. Also suppose that E: M __ IR extends to a C 2 -functional on T and that at any critical point :Z:o EM-in the sense of Definition 1.3- the form d 2E(:z:o) extends continuously to H x H with expansions for E and dE:
(3.2') (3.2")
= E(:Z:0)+~d2E(:Z:o)(:Z:-:Z:0, :Z:-:Z:o)+o(I:Z:-:Z:ol1-), (dE(:z:) , :z: - y) = d 2E(:z:o)(:Z: - :Z:o, :z: - y) + 0(1:z: - :Z:011-) E(:z:)
for all :z:, y E M such that
111 - :z:IH :::; I:z: - :Z:oIH·
53
II. Unstable minimal surfaces
Remark 3.1:
i)
In case
M
=T =H
our assumptions simply mean that
E E C 2 (H). ii)
Note that by (3.2) implicitly we assume that at any critical point = 0, Vz EM.
Zo of
E in M there holds (dE(zo), Zo - z)
Actually, E E C 1 (T) and the existence of d2 E(zo) at any critical point of E on M, together with the expansions (3.2) would suffice.
iii)
iv)
We do not assume that (3.2) holds for
z
rt.
M.
By the Riesz representation theorem and (3.2) at any critical point Zo E M there exists a bounded self-adjoint linear map a(zo): H -+ H such that
Let Ho be the kernel of a(zo), and let subs paces of H such that
H_, H+ be maximal a(zo)-invariant
(a(zo)e,e) < 0
, VeE H_, eto,
(a(zo)e,e) > 0
, VeE H+, eto.
This defines a decomposition H = H + E9 Ho E9 H _ which is orthogonal with respect to h·) and d2 E(zo), our standard decomposition of H at a critical point :to E M. (In case a(zo) has a pure point-spectrum, H+, H_ are spanned by eigenvectors of a(zo) corresponding to positive, resp. negative eigenvalues of a(zo).) Definition 3.2: Zo E M is a non-degenerate critical point of E if a(:co) is a topological isomorphism of H. In this case the Morse indez of :Co is given by
where H = H+ E9 Ho E9 H_ denotes the standard decomposition of H at :Co. We now add as a further assumption: At any critical point :Co there is a neighborhood (3.3)
{:Co}
+
U_ of 0 in H_ such that
U_ eM.
Before we state the main theorem of this section we need two more definitions .
.Definiton 3.3: Let S, T be topological spaces. attachment .,p of a handle h of type r iff
i) T
= Suh
T
arises from
S
by
54 ii)
A. The classical Plateau Problem for disc - type minimal surfaces.
"": B 1 (OjJRr
) -+
h is a homeomorphism, and also the restrictions "" laB l(O;lln
:
8B1 (Oj N)
"" IB l(O;.ln : B 1 (Oj N)
-+
-+
S n 8h.
T\S
are homeomorphisms.
Definition 3.4: Let S, T be topological spaces. Sand Tare homotopically equivalent iff there are continuous mappings f: S -+ T, 9 : T -+ S such that the composition fog: T
-+
T is homotopic to the identity on T
f: S
-+
S is homotopic to the identity on S.
and 9
0
The maps
f
and 9 in this case are homotopy equivalences.
Remark 3.5: i) Definition 3.4 defines an equivalence relation on the category of topological spaces and maps. ii) Note that certain topological properties of spaces are invariant under homotopy equivalence: If +: S -+ G(S) is a map from the category of topological spaces and maps to the category of groups and group homomorphisms which is functorial in the sense that Any f: S -+ T induces a homomorphism only on the homotopy class of f .
If f: S
-+
T, 9 : T
The identity
-+
U, then
(g 01)+
f+: G(S)
-+
G(T) depending
= g+ 0 f+.
Is: S -+ S induces an automorphism
1~: G(S) -+ G(S).
then the structure group G(S) up to automorphisms will be independent of the particular representant in the homotopy equivalence class of S. In particular, the ranks of the homology groups of S will be invariant under homotopy equivalence,cf. Djugundji [1].
For a functional E on M satisfying the above hypotheses (3.1)-(3.3) and possessing only non-degenerate critical points, let
em = I{:z: E Mlg(:z:) = 0, denote the number of critical points of E on Then we obtain the following
index(:z:)
= m}1
M with index m.
n. Unstable minimal surfaces
55
Theorem 3.6: Suppose E is a functional on M satisfying assumptions (3.1) - (3.3) and possessing only non-degenerate critical points. Let the Palais-Smale condition be satisfied for E on M. i) Then for any constant E with IE(z)1 ~ c.
c E IR there are only finitely many critical points
z of
ii) Moreover, if for a < /3 < 'Y the number f3 is the only critical value of E in [a, 'Y], and if Zlt ... , Zk are the only critical points of E with E(zj) /3 having Morse indeces rl, ... , rk, then M.., is homotopically equivalent to Ma with k handles of types rlt ... , rk disjointly attached.
=
iii) If the energies of critical points of E are uniformly bounded, then E possesses only finitely many critical points and the Morse relations hold:
,
Co
~
1,
L:(-I)'-mCm > (-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
whence
(dE(x),eo(x»)
~
~ c· (11d2 E(xo)1I +~) 2
A - xolH 2 -"2lx
~ -c-
1
Ix - zolH,
g(z) 2 ,
with a uniform constant c E JR. Finally, eo is Lipschitz continuous in the
H -norm.
Now let 1/10' 0 ~ 1/10 ~ 1, be a Lipschitz continuous function with support in U and == 1 in a neighborhood V C M-y \Ma of Xo. Let e: AI -+ T C H be a pseudo-gradient vector field as constructed in Lemma 1.8 and Lipschitz continuous in H. Remark that by assumption (3.1) the function 9 - being continuous in T, cpo Remark 1.2.ii) - is also continuous in H. Hence the construction of Lemma 1.8 conveys. Then the vector field
e(z)
= 1/Io(x)eo(z) + (1 -1/Io(z»e(x)
defined in a neighborhood of M-y \Ma will have the following properties: e is Lipschitz continuous in Hj
(3.8)
e(x) + x EM, 'V x E M-y\Ma , le(x)IH ~ c, (dE(x), e(x») ~ _min{c- 1 g 2 (z), I}
for some uniform constant c E JR.
As in the proof of Proposition 1.9 we may now construct a deformation ~ of a neighborhood of M-y \Ma by letting ~ be the unique solution to the initial value problem
t(3.9)
a
at ~(t, z)
= e(~(t, x»,
58
A. The classical Plateau Problem for disc - type minimal surfaces.
C)(O,:z:)
= :z:.
Indeed, by Lipschitz continuity of e c) exists locally. Moreover, by uniform bounded ness Ie IH ~ c and since by assumption M is also closed as a subset of H the trajectories t 1-+ c)(t,:z:) are (relatively) closed and c) can be continued globally as long as c)(t, :z:) remains in a neighborhood of M.., \Ma where e is defined. Since E is non-increasing along trajectories of c) by construction, through any :z: EM.., \Ma the solution c)(t,:z:) of (3.9 ) will be defined for
(3.10)
0< t < t(:z:)
Moreover, since 0: is regular :z: 1-+ t(:z:) is continuous.
= min{t ~ OIE(c)(t,:z:)) ~ o:}. c) meets
Ma transversally, and the time-map
In the following we replace c) by the localized flow c)a,..,: [0, oo[ x M.., by
-+
M.., given
Note that c)a,.., is continuous. This understood from now on for ease of notation let us again simply write c) instead of c)a,.., . The construction of the desired homotopy equivalence of M.., with handle of type ro attached is now completed in three stages.
Ma with a
Step 1:
=
Choose a neighborhood W of :Z:o in M of the form W ({:Z:o}+ Bp(O;H+) x Bo-(O; H_)) n M such that We V c M.., . In particular W is convex and the orthogonal projection of W onto {:Z:o} + H_ coincides with
By Lemma 1.10 there exists a constant 0
>0
such that for
W there holds g(:z:) ~ o. (3.8) then implies that for any c)(t, :z:» E Ma U W for some t or
:z: E M..,\Ma , :z: ¢ :z: E M.., either
t
E(c)(t,:z:))
= E(:z:) +
j(dE(C)(t,:z:)), e(c)(t,:z:»)dt o
~-r-t min{c- 1 0 2 ,1}.
Hence for to
= (-r-o:)
. max{l, co- 2 } the flow c) induces a homotopy equivalence
59
II. Unstable minimal surfaces
Step 2: Choose a Lipschitz continuous (in H) function V and == 1 on W. For :z: E M let
Note that
1/11. 0:::; 1/11 :::; 1, with support in
:z:' E M by (3.3). Now define a homotopy 1P : [0, to] x M
M by t 1P(t,:z:) =-:z:' + (1 to
Observe that 1P(0,.)
=
idl M
,
-+
1P(t, ')IMa
=
letting
t - ):z:. to
idl Ma for all t E [0, to] by choice of
V. Moreover, 1P(t o, W) C W_. Also remark that since e (and hence) V by construction and since W is convex we can achieve that ~
~
is linear on
([0, to], 1P(t, W)) C 1P(t, ~([O, to], W)) C ([O, to], W)
for all t E [0, to]. For this, since is strictly E-decreasing by (3.8), we can choose W for fixed V so small that no trajectory {(t,:z:)lt E [O,to]} through :z: E W can re-enter W after leaving W. Hence the composition ~ 0 1P : [0, to] x M.., -+ M.., induces a homotopy equivalence
Step 3: To conclude the proof suppose
{:Z:o} + B .. (Oj H_) C W_. For
IKo=H_ let
eE B
1
(OjIRrO) C
e=~o
,e = °
to obtain a homeomorphism 1/1 of B 1 (OjIRrO) onto h:= ([O,t o], W_) satisfying the conditions of Definition 3.3. (The time-map :z: 1-+ t(:z:) was defined in (3.10).)
o Remark S.8 If M = T = H then for any functional E E C 2 (H) assumptions (3.1)-(3.3) are trivially satisfied. Since the above constructions are purely local Theorem 3.6 also conveys to C2-functionals on Hilbert manifolds.
60
A. The classical Plateau Problem for disc - type minimal .urfaces.
4. Morse inequalities for minimal surfaces. We claim that the assumptions (3.1), (3.2) of the preceding section are satisfied if we let M be defined as in (2.2), T = HI/ 2 ,2 n C O (IR/27r) as in (2.5) , H = HI/2 ,2(IR/27r) and let E(z) be given by (2.4).
Lemma 4.1:
M is a closed convex subset of the affine Banach space {id} +
T. T is a dense subspace of H, and the topologies induced by the inclusions Me {id} + T, Me {id} + H coincide. Moreover, M is closed in {id} + H.
Proof: By Lemma 2.3 it remains to verify the last statements. Suppose by contradiction that {zm} is a sequence in M tending to Zo in H while IZm - zolT ~ C > 0 uniformly. Since the map X in (2.1) is bounded, cf. Remark 2.4, the energies E(zm) are uniformly bounded. By Proposition 2.2 Zm -+ Zo uniformly. (The case Zo = const. modulo 27r is excluded because Zo E H I / 2 ,2 and is monotone. In particular Zo E M and M is closed in { id} + Hj moreover, any relative T-neighborhood of a point Zo E M contains a relative H -neighborhood of Zo0
o Lemma 4.2: Suppose 'Y E C 4 • Then C 3 on {id} + T, and at any critical point of Definition 1.3) we have the expansion
E Zo
extends to a functional of class E M of E of M (in the sense
= E(zo) + ~ d 2E(zo)(z - zo, z - zo) + o(lz - zol1-), (dE(z), z - y) = d2E(zo)(z - Zo, z - y) + o(lz - zol1-) E(z)
for all
z, y E M
such that
Iy - zlH
:5 Iz - zolH'
Proof: The first part follows from Lemma 2.7. Moreover, by Proposition 2.9 and Theorem 1.5.1 a critical point Zo E M induces a minimal surface Xo = X(zo) E C 3 (B,IRn). Hence also Zo E C 3 • Compute, using (2.9) for {, 71 E T :
d 2E (Zo )({ , 71) (4.1)
=
J +J
d2 On X· d¢!2 'Y( zo) . { . 71 do
BB
V(dX(zo) . {). V(dX(zo)' 71) dw,
B
where
(4.2)
II. Unstable minimal surfaces
By Lemma 2.6 and the above regularity properties of continuously extends to H x H.
61 Zo, Xo
the form
d 2 E(zo)
To verify the expansion formula near zo, first note that by (1.1.2), (2.9) at a critical point Zo E M dE(zo) 0 E T*.
=
Since
E E C 3 on {id} + T we hence obtain that
= E(zo) +"21 d2 E(zo)(z - zo, z - zo) + O(lz - zol~) (dE(z), z - y) = d 2E(zo)(z - zo, z - y) + O(lz - zolf)lz - yiT' E(z)
Evidently
Iz - zol~ :S c(lz - zol~ + liz - zoll!,), Iz - YI~ :S c(lz - zol~ + Iy - zol~), and it suffices to show the following Lemma 4.3: Suppose 'Y E C r , r ~ 2, and let Zo be a critical point of E in M. Then for any p E [1,00[ there exists a constant c = c(p, zo) such that there holds for any z E M.
Proof: C 1 • Let
Since
Co
r
E C2
Proposition 2.9 and Theorem 1.5.1 imply that
= II~zolioo < 00.
By monotonicity of z E M then for any p E [1,oo[
J 2,..
liz -
zoll~l
:S(p+ 1) Co
Iz - zolP d<jJ
o
where the last estimate is a consequence of Sobolev's embedding
H l /2 ,2(IR/27r)
'--+
V(IR/27r), V P < 00.
o The proof of Lemma 4.2 may now be completed by choosing p
> 2 in Lemma 4.3.
o Verification of assumption (3.3) requires some regularity results which will be established in Chapter 5 and 6.
62
A. The classical Plateau Problem for disc - type minimal.urfaces.
Lemma 4.4: Suppose 'Y E C r , r ~ 5, and let Zo E M be critical for E on M, H H+ ffi Ho ffi H _ the standard decomposition of H at zoo Then Ho, H_ are finite dimensional and regular: Ho ffi H_ C C 1 (IR/27r).
=
Cpo Proposition 5.6. Recall our representation TidG
= span{l,sin¢,cos¢}
of the tangent space to the conformal group G at id , acting on BE!:!! IR/27r . By conformal in variance of D for any 9 E G any z EM, X X(z) E C(r) we have: E(z 0 g) D(X 0 g) D(X) E(z).
=
=
Hence at any critical point
d 2 E(z)
=
Zo E M
d (dE(z), d¢ z·
(4.3)
=
e) = 0, "t eE Tid G ,
(d~z .e,17) = 0, "t eE TidG,
"tTl E H,
and the non-degeneracy condition of Theorem 3.6 cannot hold for where the conformal group G is acting. We may attempt to normalize admissible functions z e;k) 2;k, k E ZZ . Thus we consider
=
M* C {id} where T*
+ T*
E on any space
z by a three-point-condition
,
= {e E Tie C~k) = 0,
k E ZZ} .
However, since Hl/2,2 fails to embed continuously into CO , taking the closure of T* in H 1 / 2,2 we reobtain the full space H and the problem of degeneracy remains. For this reason we choose to replace the pointwise constraints by integral constraints and work in the class instead, where
J 2,..
Ht =
{e E H I e17 d¢ = 0,
"t17 E 1"id G } .
o Clearly, assumption (3.1) will be satisfied for the triple (Mt, Tt, Ht). Moreover, by Proposition 2.9 any critical point Zo E Mt of E will also be critical for E on M; by Lemma 4.2 therefore d 2 E(zo) extends to Ht x Ht C H x H and assumption (3.2) will be satisfied.
63
II. Unstable minimal surfaces
Lemma 4.5: Suppose 'Y E cr, r > 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] d
0
~
for any h-j:O, any 211"- periodic
0
and 1].
Moreover, we need the following lemma. If X E H 1 ,2(Bj JR.n) is harmonic, and iffor all h-j:O there holds
Lemma 5.2:
!
IV8,.XI 2 dw
5: C < 00
B
uniformly in h, then X E H 2,2(Bj JR.n).
Proof:
By weak compactness of L2, there exists f E L2 such that
h
weakly in L2 for some sequence C;;"(Bj JR.n)
J
J
B
B
V8,.X .rpdw = -
-+
0. On the other hand, for any
VX8_ h rpdw
-+ -
Jvx B
:
rp E
67
II. Unstable minim.al surfaces
as h - 0, whence
= V fJ4JfJ X
f
2
E L (B).
But then by harmonicity of X ,on B \ B i / 2 we can estimate
which is integrable over the annulus B\B i / 2 (0). Since follows that V 2 X E L2(B), i.e. X E H 2,2(BjDl").
X is analytic in B, it
o Proof of Proposition 5.1:
In (5.1) choose £
Y =:z: + £fJ_hfJh :z: = :z: + h2 [:1:+ - 2:1: = (1 _ Note that :z:+,:1:_ E M, 0:::; 2£ :::; h 2 • Moreover, since periodicity
:I:
~~):z: + ~~
2,..
(:z:+; :1:- ) .
whence by convexity of
E Mt also
+ :z:_]
M
we have
y E Mt . Indeed, denote
e = :z: -
y E M
id
j
then by
2,..
I e± d4J I ed4J I e± 4J d4J = ! e =
o
= 0,
0
2,..
2,..
sin(4J =F
sin
o
h) d4J
0
2,..
2,..
= cos hie
sin 4J
d4J =F sin hie cos 4J d4J
o
and likewise
=0 ,
0 2,..
I e±
cos 4J
d4J
=0 .
o
I.e. (5.2)
y - id
= (1 -
~)
e+ pe+ + pe- E Tt
I lJB
On the other hand, on fJB
. From (5.1) we then obtain that
fJnX d~ -y(:z:) . fJ_hfJh
:I:
do 2:
o.
if
68
A. The classical Plateau Problem for disc - type minimal.urfaces.
and therefore by the product rule
8_ h 8h X
=d¢>d 'Y(:r:)8_ h8h:r:+
-(* 7d~'7("W) .a••_ *1!d~'7(''')dz''d')
(5.4)
+8_. ( Now extend 8h:r:, etc. to hence infer that:
+
f
B
harmonically. * Integrating by parts in (5.2) we
*f f
x+ x'
V 8h X V (
B
~ c(e).
f
.
x
) dd;2 1'( :r:")d:r:" d:r:'
dw
x
(IVXI 2 + IV:r:12) (1 + 18h:r:1 2)
B
+ I V8h X 12 dw. But on 8B, by (5.3)
Z+
d () -1' :r: . 8h X - -1 h
d¢>
ff x
:&'
2 d¢>2
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
<1
the function
Indeed, a.e. on aB=[0,211'] we have
and -lp:c ~ 0 a.e. while by monotonicity of :c and our choice of second term is non-negative. Now pretend that inequality (5.1) is valid for y. Then we obtain
f a"x :4> liB
"Y(:C)(:C -
:C o
)r 2 do ::; O.
r
also the
70
Let
A. The classical Plateau Problem for disc - type minimal surfaces.
Xo
= l' (:Z:o) E r.
(5.5)
Then on 8B
X - Xo
., .,
= d~ l' (:z:)(:z: -
:Z:o) - / / 2:0
d~21'(:z:I/)d:z:l/d:z:/.
x'
Upon an integration by parts we hence obtain that for any pre - assigned 6> 0 with a constant C6 there holds:
+6
/ IVXI 2T2dw + C6
/
B
B
IVTI 21X - Xol2dw.
After another integration by parts
~ c· /
IVXIIV:z:II:z: - :Z:01T2 + IVXIIVTII:z: - :Z:012Tdw
B
~ cll:z: -
:Z:o ilL 00 (B3r(wo» / (lVXI2
+ IV:z:12) T2 + I:z: -
:Z:012IVTI 2dw.
B
Thus, for 0 < r < ro sufficiently small (depending on continuity of :z:) we obtain the estimate
~ 6/
/ IVXI 2T2dw (5.6)
B
6 and the modulus of
(IVXI 2 + IV:z:12) T2dw
B
+ c/
(IX - Xol2
+ I:z: -
:Z:012) IVTI 2 dw.
B
Nowextend :z: harmonically to B n B3r(WO) . Note that
(V:Z:)T
= V«(:z: -
:Z:o)T) - (:z: - :Z:o)VT
and that
.6.«(:z: - :Z:o)T) = 2V:z: VT + (:z: - :Z:o).6.T =: f has support only on B3r(Wo)\B2r(Wo). Let
d~ 1'(:Z:) • (X (5.7) (z -
.,
Xo)
+
.,
JJ
:f/J 1'(:Z:) . :f/J 2 1'(:z:II)dzll dz l
So . '
Zo)T
2
= ---------.:':.:::dd....:f/J:...1'-(:z:-)-:
71
II. Unstable miniInal surfaces
on 8B U (B n 8B3r (w o
».
Recall the variational characterization
D(cp) :::; D(t/J)-
=
=
for the solution cp (z - zo)r of the eqution b.cp f in boundary data CPo. Upon inserting t/J = CPo this implies that
f IVzl2r 2dw :::; cf B
IVXI2r 2dw
f
+ c (IX - Xol2 + Iz - zol2) (lVr12 + Ib.rl) dw
B
+ cIIIX - Xol + Iz -
B n B 3r (w o ) with
B zolllioo(BnB3r(1Do»
f (lVr12 + IVz 12r2)
dw.
B
r > 0, depending on the
Together with (5.6) this estimate (for sufficiently small modulus of continuity of z) yields that
f
(IVXI 2+ IVzI2)
dw:::; cr- 2
Br(1Do)nB
f
(IX -
Xol2 + Iz - zol2) dw.
(BnBar(1Do )\B2r (wo»
The right hand side in this expression can now be further estimated by means of a Poincare - Sobolev - type inequality, cpo Lemma 5.5 below: (We omit writing Wo in the sequel.)
f
f +c( f +( f
(IVXI 2+ IVzI2) dw+
(IX - Xol2 + Iz - zol2) dw:::; cr 2
Bn(Bar \Br)
Bn(B3r \Br)
(X-Xo)dOr
8BnB2r\Br
8BnB2r\Br
By choice of
Zo
! 8BnB2r\Br
(z-zo)do=O,
(Z-Zo)dOr·
72
A. The classical Plateau Problem for disc - type minimal surfaces.
while by (5.5) and an application of the trace theorem:
(X - Xo)do
I
=
8BnB2r\Br
'"
J JJd~2 J J IV;Z; 12dW+C( J J J + IV;z;12)
+
",'
1'( ;z;")d;z;" d;z;' do
8BnB2r \Br "'0 "'0
8BnB2r\Br ~
cr
=cr
(;z;-;z;o)dof
8BnB2r\Br
BnB2r\Br
BnB2r\Br
I.e. if we let
~(r) =
(IVXI2
dw,
BnBr(wo)
we obtain the difference inequality
Cl ~(r)
We now use Widman's [1] "hole filling" trick: Add divide by 1 + Cl to obtain
to this inequality, and
Cl
~(r) ~ --~(3r).
1 +C1
Let 0 results
= 1 ~l C1.
By iteration, for any r
>
0, k E IN such that
~(r) ~ 0k~(3kr).
Finally, for any r E]O, ro[ there is exactly one k E IN such that
ro E]3 k - 1r, 3k r]. Clearly, k is the smallest integer k
> In (~)/ln3,
whence ~(r) ~ 0- 1 (
We may now choose J.'
r ) In 0/1n3 :
= -ln0/1n3 > 0, ~(ro)
Co
~(ro).
= 0r'"o
3kr ~ ro there
73
II. Unstable minimal surfaces
to complete the proof of the lemma, if y E Mt. To achieve the normalization y E Mt we modify z near three points eiq,j, j 1,2,3 simultaneously where the angles tPj differ by approximately Then, if we let
=
~(r) =
J
sup woEOB
= 2; .
Wj
(IVXI 2+ IVzI2) dw
BnB.(wo)
our above reasoning leads us to an estimate ~(r)
with a fixed constant iteration as before.
e<1
$
e
~(3r)
for all r $ ro ,and the desired conclusion follows by
o Proof of Proposition 5.1
X(W)
(completed):
Extend X to JR2 by reflection
= X C~2)'
V W ¢ B.
Choose r E]O, 1/2] and let T E C:'(B2r(O» satisfy T == 1 on Br(O). Cover B by balls of radius r in such a way that at any point wEB at most k balls of the cover intersect, k independent of r. Let the balls Bi of this cover be centered at points Wi, and let Ti(W) == T(W - Wi). Then c- 1
J(I VahX I2+ Ivahzl 2)
dw
$~
J(IVXI 2+ IVzI2) (lahX l2+ lah Zl2)
Tl dw.
• n2
B
=
Moreover, by Lemma 5.4 the function "" (IVXI 2+ IVzI2) satisfies a Money growth condition as in Lemma 5.3. Applying Lemma 5.3 with this function "" and I()i = ahXTi, resp. I()i = ahzr; E H.!·2(B 2r (Wi» we obtain that
J (IVXI 2+ IVzI2) (lahxl 2+ lahZl2) J (IVXI 2+ IVzI2) J (lvahxl 2+ Ivahzl 2) + J IVXI2+ IVzI2) J (lahxl 2+ lahZn IVTi12dw. Tl dw
$ crl'/2
dw
B 2 (0)
crl'/2
2r (wi)
dw
B2(0)
Summing over holds:
dw
B
B2r(tDi)
i we may infer that with a constant
J(lvahxl 2+ Ivahzl 2)
dw
$clrl'/2
Cl
independent of r there
J (Ivahxl2+ Ivahzl 2)
~~
B
+cr- 2
f
B2(0)
(l8"XI 2 + 18h z 12 ) dw.
dw
+
74
A. The classical Plateau Problem (or disc - type minimal surface..
However, bounds for X and its derivatives on B imply similar bounds on B 2 (0). Moreover O,..X -+ -Jq;X strongly in L2. Hence we deduce that
! (IVo,..XI + IVo,..:c1 2
2)
dw
~ c1r/S/ 2
B
! (lVo,..XI + IVo,..:c1 2
2)
+ cr- 2
dw
B
and for sufficiently small r
>0
there results a uniform a-priori bound
! (IVo,..XI + IVo,..:c1 2
2)
dw
~c
B
independent of h. By Lemma 5.2 X E H2,2(BjJR"), and the proof is complete. []
For completeness, we give a proof of Poincare's inequality that we have used in the proof of Lemma 5.4:
Lemma 5.5:
Let
3r
< 1, Wo E oB, Br
= Br(w
o ),
etc.
B n
G
Set
(B3r \Br ), 8=oBnB2r\Br' Then for any tpEH 1.2(G) there holds
with a constant c independent of r, w o , and
Proof:
tp.
Suppose by contradiction that for a sequence {tpm} in
Then {tpm} is bounded in
H1,2(G) and we may assume that
tpm
H1,2(G)
~ tp weakly
H1,2(G). By the Rellich-Kondrakov theorem also tpm -+ tp strongly in L2(G) and in L1(8). Moreover, Vtpm -+ 0 strongly in L2(G), J tpmdo -+
in
0,
whence
J Itpl2dw = 1.
tp
==
const
= O.
5
But by strong
L2-convergence we must have
G
The contradiction proves the estimate for fixed of rand Wo is seen by scaling.
r. That
c is in fact independent []
Proposition 5.6: point of E on
Suppose "Y E or, r ~ 5, and let :1:0 E M be a critical M corresponding to a minimal surface Xo = X(:C o ) E
75
II. Unstable minimal.urfaces
spanning r. Let H = H+ EB Ho EB H_ be the standard decomposition of H = H l/ 2,2(m/27f) induced by Jl E(zo). Then dim (Ho EB H_) < 00 and HoEBH_Ce l .
c(r)
Proof: Let H Xo. Recall that
= dX(zo)(H)
be the space of H l ,2- vectorfields "tangent" to
d2 D(Xo)(e, 7)= /V{Vr,dW+ / (5.8)
B
I>X o(r)e7)
Id~XoldO
8B
=d2 E(zo)(e,71) for all ting
e,71 E H,{ = dX(zo).e,7) = dX(zo)· 71 E H. d2 D(Xo) induces a splitH = H+ EB Ho EB H_, and since dX(zo) = h (~'Y(zo).) is a topological H onto H from (5.8) it is clear that also dX(zo)IH+: H+ -+ dX(zo)IHo: Ho -+ Ho, and dX(zo)IH_: H_ -+ H_ are topological
isomorphism of
H+,
isomorphisms. It hence suffices to show that Garding type inequality
(5.9)
dim (Ho
d2 D(Xo)({,e)
~/
+ H_) < 00.
IVel 2dw -
B
c/
But this follows from the
1{1 2do
8B
and a standard argument: If {em} is an orthonormal system in implies that
(5.10)
1 = / l{ml 2+ IVeml2dw ~ B
c/ 8B
Ho EB H_, (5.9)
leml2do + / leml2dw B
for all m. But the embeddings H l ,2(B;mn ) '---+ L2(B;mn) resp. '---+ L2(BB; m) are compact. Hence {em} must be finite, or (5.10) would eventually be violated for a sequence {em} weakly tending to 0 in H l ,2(B; mn). To obtain regularity of an element e E Ho EB H_, we once again use difference quotients B_hBhe, etc., as introduced at the beginning of this chapter.
(5.11) and
Extend e, Bhe, may think of
e
etc. harmonically into B. (Since e is 2?r-periodic, we as a member of H l / 2 ,Z(BB), as well.) Note that in the
76
A. The classical Plateau Problem for disc - type minimal surfaces.
H l / 2 ,2-scalar product induced by the restriction
*
H l / 2 ,2(8BjIRn )
(e,17h/2 =
JVe V17
H l ,2(Bj IRn) 3 X
-+
XI8B E
dw
B
where
e,17
denote the harmonic extensions of
e,17
to
B,
we have
(5.12)
(5.13) where by Sobolev's embedding theorem and Young's inequality Rl
= d2D(Xo) (i, 8_h(8h(dX(:z:o)) . e+) + 8_h(dX(:z:o)) . (8he)- )
JIV8hfi (Ie+ 1+ IVe+1) + Ivfl (18he-1 + IV8he-1) + J18hflle+ I + IfI le+ I + lil18he-ldo JIV8hfl 2+ IV8hel 2dw + S c·
dw
B
c
8B
Sf
C(f).
B
C(f) may depend on the Hl,2-norm of f, resp. the Hl/2 ,2-norm of f. The estimate for Rl requires C 3 -estimates for Xo. But by (5.8)
d 2D(Xo)(f, 8-h 8hf)
(5.14)
=-
JIV8hfl dw 2
R2
B
where
R2
J SC J
= /exo(r)18hfI2Id~Xol do + 8B
J
8h (/exo(r)
Id~XOI) f· 8ddw
8B
18hfl2do + c
Sf
JIV8hfl 2dw +
8B
B
Again we require
Xo E C 3 (Bj IRn).
C(f).
Moreover, note that by (5.11)
!
/V8he/2dw S c
B
*
cpo the proof of Lemma 2.5
! B
/V8h{/2dw + C.
e,
and
77
II. Unstable minimal surfaces
Finally, combining (5.13), (5.14)
Jlvoh{1 2 dW < IRll B
+ IR21
:::; f / IVoh{1 2 dw + C(f), B
and it follows that
/ IVOhfl2dw:::; C(f) B
uniformly in
h>
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 ~
= IR/21f
where
q
VA:
E IN,
L: = v}. VA:
A:=1
In order for
zED" to be monotone, necessarily each
VA:
must be even.
We may think of D as a subset of our space M, introduced in (2.2), and at each zED" the tangent space T",D" C H, d. Section 4. Finally let
be the space of harmonic surfaces in 1R" decomposed into bundles T/~ consisting of harmonic surfaces with p interior and q boundary branch points of orders
84
A. The classical Plateau Problem Cor disc - type minimal surfaces.
"1, ... , ", such that the total order of interior, resp. boundary
AI, ..• , -\p resp. branching is
p
,
:L: Ai = A, :L:"" =
i=l
II.
"=1
More precisely, to each 'Y E A, z E Dv associate a harmonic surface X = h{ 'Y 0 z) with XI"B 'Y 0 z. Consider the holomorphic vector function of w u + iv
=
=
(6.2)
F{w)
== Xu
ax.
- iX.. =
Taking the square of F
(6.3) we obtain another holomorphic function over B. Now let
g (w - e'"t" P,
p ,
,.,~
= { h,z) E A
x Dvl F
= }l{w -
Wi)Ai
Pf-O inB, Wi
where the holomorphic function preceding formulas.
F
EB, Ai EIN, ~ Ai = A}
is associated to the pair
('Y, z)
by the
The topologies on A, D v , ,.,~ are certain Hilbert space topologies defined with reference to 'Y,0,{¢,,}, {Wi}, and P resp. Define the bundle projection
11:,.,-+A by letting 11~:,.,~ -+ A be the map 11~{'Y, z) = 'Y.
Also introduce the conformality operators K~ from ,.,~ into the space
of hoiomorphic functions having (at least) 2A interior zeros Wi with multiplicities 2Ai and 2" zeroes e'" E 8B with multplicities 2111:, by letting
be given by (6.3).
85
II. Unstable minimal surfaces
The set of minimal surfaces M in TJ now is stratified M=
u
M~ v
into subsets of minimal surfaces of a prescribed branching type. In the appropriate Hilbert space topologies, the operator K;: 17: -+ Y,,~ differentiable. Moreover, its differential at a minimal surface h,:z:) E 17~ surjective.
IS
is
By the implicit function theorem this implies the first part of the following "Index Theorem" of Bohme and Tromba. (Cp. Bohme and Tromba [1, Theorem 3.39]. The case v,.W was completed by U. Thiel [2].)
Theorem 6.11:
M~ is a differentiable submanifold of 17~. The restriction
is Fredholm (i.e. is differentiable and at any (1", :z:) E M~, d (II~ IM~) has a R of finite codimension and a finite dimensional kernel N) of closed range Fredholm-index dim(N) - codim(R)
= 2(2 -
n)A
+ (2 -
n)v + 2p + q + 3.
Remark 6.12: i) If we normalize with respect to the conformal group on B the Fredholm index of the projection map becomes 2(2 - n)A + (2 - n)v+ 2p+ q.
ii) Note that by the Plateau boundary conditon boundary branch points have even multiplicities Vk ~ 2. Hence with such a normalization for n ~ 3 the index of II~ IM~ at a minimal surface X associated with h, :z:) E M~ is non-positive v and it may equal zero only in case
v
A = v = 0, Aj ::; 1,
= 0,
if n if n
~ 4
= 3.
iii) By the Sard-Smale Theorem almost all 1" E A are regular values for all the (countably many) projections II~IM~' Denote this set of regular values 1" E A by
d
(II~ IM~)
.A.
By ii) necessarily the index of II~ IM~ at
also is injective at
1" ,i.e. each
1" E A equals 0, and
M~ is transversal to the fibre over 1" E A are isolated and stable under
1" E.A . In particular, minimal surfaces over perturbations of 1" in any manifold M~ (with
A, v fixed). Moreover, by the
86
A. The claasical Plateau Problem for disc - type minimal surfaces.
a-priori estimates of Theorem 1.5.1 the set of normalized minimal surfaces spanning a wire 'Y is compact. However, different sheets M~::: cannot accumulate at a regular minimal surface X (X would have to have infinite order of branching). Our Remark 6.12 translates into the following generic finitene, and ,tability result of Bohme-Tromba [1, Theorem 4.14]: Theorem 6.13: There exists a dense open set of curves A c A such that any A bounds only finitely many minimal surfaces. These surfaces are stable under perturbations of 'Y. Moreover, they are immersed over B if n ~ 4, resp. may possess at most simple interior branch points, if n = 3.
'Y E
Theorem 6.13 also provides a (partial) answer to the question whether for a curve r the non-degeneracy condition in our Theorem 4.6 is satisfied. Corollary 6.14: Let n ~ 4, and let A be as in Theorem 6.13. Then any 'Y E A spans only minimal surfaces which are non-degenerate in the sense of Definition 3.2. Proof: h( 'Y 0 z)
If n ~ 4, 'Y E of branching type
isomorphism of T-y,,,,M~
=
A
bounds only immersed minimal surfaces A= 1/ = 0, and such that d (II~ IMg)
ker(dK~)
X
=
IS
an
onto T-yA.
In particular, the restriction d",K~ of dKg to the tangent space T",'Do ~ Ht at z in the fibre of normalized surfaces over 'Y is injective:
(6.4) We rewrite this expression in a more convenient way: Note that if X is harmonic also the function
is holomorphic over B. Moreover, Kg('Y,z) = 8X 2 ,
whence (6.4) implies that also the holomorphic functions
(where d", again denotes the derivative with respect to z). In fact, for all such Y the real and imaginary parts separately cannot vanish identically. Else by the Cauchy-Riemann equations Y == const YeO) 0, because Y contains the factor w 2 •
=
Finally, by harmonicity of im(Y) we conclude that im (ds9('Y, z) . {)
= 2 d", (X(z)... X(z).) . e#O
on 8B
=
87
II. Unstable minimal surface.
for all
e E T,';D o, e=l=O,
J = Jo..
and we can find
(d", (X(z),. . X(z)",).
X·
=d2 E(z) I.e.
E T",Vo such that
e)· ~ d~ =
DB
DB
~
d22'Y(Z)~Z·e· ~d~+ d~
d~
J (h (~'Y(z) 0..
DB
d~
.e)) . ~'Y(z). d¢
~z. ~d~ d~
(e, :~ z· ~) =1=0.
X is non-degenerate, and the proof is complete. []
The representation of minimal surfaces by smooth submanifolds M~ in the bundle 1/ also allowed Tromba to apply a variant of degree theory to the Plateau problem. In this way, in 1982, he was able to give the first proof of the "last" Morse inequality using a geometric notion of "index" for minimal surfaces spanning a generic curve r in Ilt', n 2: 3, cf. Tromba [2]: Theorem 6.15 : For n 2: 3 let A be as in the statement of Theorem 6.13. Then for any rcA the Morse relation holds:
L(-I)i(X) = 1, x where the sum extends over all the (finitely many) minimal surfaces. X spanning and i(X) equals the Morse index of X, in case X is non-degenerate, as defined in Section 4.
r,
However, the "intermediate" Morse inequalities cannot be obtained from degree arguments, and a different approach as outlined in Section 4 was needed to complete the analysis.
B. Surfaces of prescribed constant
mean curvature
III. The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in m? 1. The variational problem. Let r be a Jordan curve in IIf. In part A we studied minimal surfaces spanned by r, and we observed that any solution X to the parametric Plateau problem (1.1.1) - (1.1.3) parametrizes a surface of vanishing mean curvature (away from branch points where VX(w) 0).
=
A natural generalization of the classical Plateau problem therefore is the following question: Given r c IIf, H E JR, is there a surface X with mean curvature H (for short" H-surface") spanning r? We restrict ourselves to surfaces of the type of the disc B. Introducing isothermal coordinates over B on such a surface X we derive the parametric form of the Plateau problem for surfaces of constant mean curvature:
(1.1) (1.2) (1.3)
b.X
= 2H Xu 1\ X..
in B
!Xu !2_!X.. !2=0=Xu ·X. X!8B : 8B
=
-+
r
In
B
is a parametrization of
r.
Here a 1\ b (a 2 bl - b2 a l , alb l - bla l , a l b2 - bl a 2 ) denotes the exterior product of a (a\a2,~l), b b\b\bl ) E IIf. Formally, these are the Euler-Lagrange equations corresponding to the functional
=
(1.4)
=
DH(X)
= D(X) + 2HV(X),
where
(1.5)
V(X)
= 1/3 /
Xu
1\ X ... Xdw
B
denotes the "volume" of X. In fact, V(X) measure the algebraic volume enclosed in the cone segment consisting of all lines joining points X(w) on X with the origin. This is immediate for surfaces whose coordinates are linear functions: X( u, v) = au+bv+c, a, b, c E IIf. For smooth surfaces this geometric interpretation of the volume may be obtained by approximation of X with polyhedral surfaces. Likewise, we may also regard V(X) as the algebraic volume bounded by X and the surface Xo consisting of all line segments joining points on r = X(8B) with JJ..~.iJDD..
92
B. Surfaces of prescribed constant mean curvature
However, we need not be too concerned about the geometric meaning of simply take formula (1.5) as a definition.
V, and
Remark 1.1: i) V(X) is well-defined and tri-linear, in particular analytic on the space H 1 ,2 n L OC (BjlR. 3 ). ii)
(1.6)
For any X, tp,,,p E H 1 ,2 n LOC(Bj lR. 3 ) there holds the expansion
V(X
+ tp)
= V(X) + (dV(X), tp) + 1/2 d 2 V(X)( tp, tp) + V( tp),
where (1.7)
d'V(X)(~,,p) =1/3 ~ (~. A,p.+,p. ~.). Xdw A
(1.8)
on
+ [(~. A X. + X. A ~.) . ,p + (,p. A X. + X. A ,p.) . ~dw].
iii) If p, tp,,,p E C 2 (Bj lR.3 ) and at least one function p, tp, or "p vanishes BB, then by partial integration and antisymmerty a /\ b -b /\ a:
=
J
(tpu /\ "pv
+ "pu /\ tpv) . pdw =
B
J =J =J(pu =-
tp /\"pv . Pu
(1.9)
J
+ "pu /\ tp . Pv dw +
B
(tp /\ u"pv
+ v"pu /\ tp) . pdo
aB
(pu /\ "pv
J =J
+ "pu /\ Pv) . tpdw +
B
tp /\ :¢!"p . p do
aB
/\"pv
+"pu /\ Pv)' tp dw
B
(tpu /\ Pv
+ Pu/\ tpv)'"p dw.
B
=
Here, I¢"p u"pv - v"pu as usual denotes the derivative of "p in" angular", i.e. counter - clockwise tangent direction along BB. Hence, if either tp,,,p or P == 0 on BB, the boundary integral vanishes. In particular, if X E C 2 (B;lR. 3 ), tp,,,p E Cg"(B;lR.3 ) there holds
(1.10')
(dV(X), tp)
J =J =
Xu. /\ Xv . tpdw ,
B
(1.10")
d 2 V(X)(tp,,,p)
(tpu /\"pv +"p" /\ tp.,). Xdw.
B
m.
93
The existence of surfaces of prescribed constant mean curvature
iv) V is invariant under orientation-preserving reparametrizations of X: Let X E H 1,2 n L OO (BjJR3), and let 9 E C 1(BjJR2) be a diffeomorphism of B onto a domain iJ with det(dg) = g~g~ - g;g; > 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)(
i#
:s c(D(X)D(ip)D(1jJ))1/2. Finally, also V extends to a trilinear functional in the components of (ipl,ip2,ip3) E H;,2(BjJR3) by the estimate
IV( ip) 12
= IV( ipl, ip2, ip3) 12 :s 3!7r D( ip)3,
and (2.1) follows. Let us now establish the continuity properties asserted in (2.4), (2.7), (2.8). Proof of (2.4):
By (1.9) for any ip E C~(BjJR3)
(dV(Xm)' ip)
=/
Xmu /\ X mll • ipdw
B
=1/2 / B
(ipu /\ X mll
+ Xmu /\ ipll)' Xmdw.
ip
=
97
ill. The existence of surfaces of prescribed constant mean curvature
Note that by compactness of the embedding H 1,2(BjJR3) <-t L 2(BjJR3) Xm-+ strongly in L2 while the products ipu /\ Xmv etc. remain uniformly bounded in L2. Hence with error o( 1) -+ o( m -+ (0)
X
(dV(Xm)' ip)
= 1/2
I
(ipu /\ Xmv
+ Xmu
/\ ipv)· X dw
+ 0(1)
B
and by weak convergence Xm ~ X in H 1 ,2 the right hand side tends to
1/2
I
(ipu /\ Xv /\ Xu /\ ipv) . X dw
= (dV(X), ip)
B
as required. This proves (2.4) for smooth ip. The general case follows by density of ego(Bj JR3) in H;,2(BjJR3) and the uniform boundedness of the family {dV(Xm)} of linear functionals on H;,2(BjJR3) which results as a consequence of (2.3) and the boundedness of weakly convergent sequences in H 1,2(Bj JR3).
Proof of (2.7), (2.8):
Similarly, for
d 2V(Xm )(ip, 'I/J)
=
I
ip, 'I/J E
(ipu /\ 'l/Jv
B
-+
I
ego (Bj JR3)
+ 'l/Ju /\ ipv)· Xm
(ipu /\ 'I/J"
dw
+ 'l/Ju /\
dw
B
=dV(X)(ip, 'I/J), and the general case follows using (2.6). Likewise, for X E e1(Bj JR3),
W
ip,
'l/Jm ~ 'I/J in H;,2(Bj JR3) by arguments as in the proof of (2.4)
d 2 V(X)(ipm' 'l/Jm) =
I I
(ipmu /\ Xv
+ Xu /\ ipmv) . 'l/Jm dw
B
-+
(ipu /\ Xv
+ Xu/\ ipv)· 'l/Jdw
B
=d2 V(X)( ip, 'I/J),
and the general case follows by density of e1(Bj ml) in The proof of iv) is slightly more delicate. Let
Hl,2(Bj JR3) and (2.6).
X m , X E C(r) and assume that
Xm !£. X weakly on H 1 ,2(BjJR3) and uniformly on B.
98
B. Surfaces of prescribed constant mean curvature
Then upon integrating by parts we find
=2
6[V(Xm) - V(X)]
J
X m,.
Xmv . Xm - X,.
1\
1\
Xv . X dw
B
=/
«Xm - X),.
1\
+ X m,. 1\ (Xm -
Xmv
Xlv) . Xm dw
B
+/
«Xm - X),.
1\
Xv
+ Xu 1\ (Xm -
X)v)· Xmdw
B
+2/
Xu
1\
Xv . (Xm - X) dw
B
=/
(Xm,.
1\
+ Xlv + (Xm + X)u 1\ Xmv + 2Xu 1\ Xv)· (Xm
(Xm
- X) dw
B
+/
(Xm - X)
1\
[(u. Xmv -
V·
Xmu) + (u. Xv
-
V·
Xu)]· Xm
do
DB
~c·
(D(Xm)
+ D(X)) ·IIXm- XIILOO(B)+
+ c·IIXm- XIILOO(DB) /
:¢
I
DB
:¢ xl
Xm I+ I
do
-+0,
since
IIXmIILoo(DB) ~ c(r), f I/q;Xml do = f l/q;xl do = L(r) < DB
00.
This
DB
proves (2.9). (2.10) follows since by (1.9) if tpm ~ tp E HJ,2(BjJR3):
2[(dV(Xm), tpm) - (dV(X), tp)]
J +J =J +J =
=2
J
(Xmu
1\
X m" . tpm - Xu
1\
X" . tp) dw
B
«Xm - X)u
1\
(Xm
+ X)" + (Xm + X)u 1\ (Xm
- X),,). tpm dw
B
2X"
1\
X" . (tpm - tp)dw
B
(tpm"
1\
(Xm
+ X)" + (Xm + X)" 1\ tpm,,) . (Xm -
X)dw
B
(X"
1\
(tpm - tp)"
+ (tpm
- tp)"
1\
X,,)· Xdw
B
-+ 0
(m-+oo).
A similar reasoning shows (2.11). I]
m.
The existence of surfaces of prescribed constant mean curvature
Remark 1.1 and Theorem 2.3 imply the following Remark 2.4:
i)
DH extends to an analytic functional on
(H 1,2 n LOO(Bj JR3))+H~,2(Bj JR3) with (2.12)
DH(X
(2.13)
ii)
+ rp) =DH(X) + (dDH(X), rp} + (1/2 )d2 DH(X)(rp, rp) + 2H V(rp) =DH(X) + DH(rp) + (dDH(X), rp} + H d2 V(X)(rp, rp),
X is conformal and weakly solves (1.1) iff
= °E (H~,2(BjJR3)r , ~ DH(X (id + f'T)-l)!€=O = :€ D(X
dDH(X)
0
i.e. iff X is stationary for
0
(id + €'T)-lbo
= 0,
V
'T
E C1(Bj JR2),
DH on C(r) in the sense of Lemma 1.2.2.
99
100
B. Surfaces of prescribed constant mean curvature
3. "Small" solutions. The existence of relative minimizers of DH in C(r) has been obtained under various geometric conditions relating rand Hand under certain a priori restrictions on admissible solution surfaces. By Remark 2.4. ii) a relative minimizer X E C(r) of DH weakly solves (1.1) -(1.3). Later we shall see that X is as regular as the data permit and hence also solves (1.1) - (1.3) classically. Let us first state the following result of Hildebrandt [2] which improves earlier results of Heinz [1] and Werner [1]: Theorem 3.1: HEm. satisfy
Let
r
c
BR(O)
c
IIf be a rectifiable Jordan curve, and let
IHIR ~ 1.
(3.1)
Then there exists a solution X E C(r) to (1.1) -(1.3) characterized by the conditons
(3.2)
IIX IIL oo
(3.3)
DH(X) =min{DHIX E C(r),
~R,
IIXllLoo ~ I~I ~ co} .
Moreover, if IHIR < 1, X is a relative minimum of DH in to the topology of H 1 ,2 n LOO(B; m. 3 ).
C(r) with respect
Heinz' non-existence result Theorem 4.1 below shows that the condition (3.1) cannot be improved when r is a planar circle. For lengthy curves, however, Theorem 3.4 below by Wente and Steffen may give better results. Let CH
Note that
Cw:j;0 for
Lemma 3.2: Proof:
Hence for X E CH
(3.5)
~
1
jHj}.
HEm. satisfying (3.1).
DH is coercive on CH with respect to the H 1 ,2(B; m.3 )-norm.
Note that for
(3.4)
= {X E C(r)IIIXIILoo
X E CH there holds
12~ Xu 1\ X"
.xl ~ ~IHIIIXIILOO IXuIIX,,1 ~ ~IVXI2,
a.e. on B.
101
III. The existence of surfaces of prescribed constant mean curvature
and coerciveness of DH follows as in example 1.3.5.
o Lemma 3.3: convergence in Proof:
DH is lower semi-continuous on
CH with respect to weak
H 1 ,2(BjJR3). Xm ~ X E CH
Suppose
weakly in H 1 ,2(Bj JR3).
By the Rellich-
L 2(BjJR3) and a.e. on
Kondrakov-Theorem Xm -+ X strongly in Egorov's theorem then, for any 6 > 0 there is a set IB61 < 6 such that Xm -+ X uniformly on B\B 6.
B6 C B
B. By of measure
Hence by (3.4) and example 1.3.4. ii)
! ! !
IVXml2dw + (2H/3)
DH(Xm ) =1/2
B
?,1/2
! B
IVXml2dw + (2H/3)
B\B 6
?,1/2
Xmu /\ X m.. . Xmdw
! !
Xmu /\ X m .. . Xmdw
B\B 6
IVXI 2dw+(2H/3)
X,,/\X... Xdw-o(l)
B\B 6
B\B 6
=DH(X) - (1/2
!
IVXI 2dw + (2H/3)
B6
where 0(1)
-+
0 as
m
-+ 00,
!
Xu /\ X .. . XdW) - 0(1)
B6
for any fixed 6> O.
By absolute continuity of the Lebesgue integral
1/2
!
IVXI 2dw + (2H/3)
B6
!
Xu /\ X .. . Xdw
-+
0 (6
-+
0),
B6
and we obtain that
m-+
00
o
as desired.
Proof of Theorem 3.1:
Also let
By conformal invariance of D H
:
102
B. Surfaces of prescribed constant mean curvature
Note that by Proposition 1.4.5 C1 is weakly closed in H I ,2(B;JR.3). Moreover, DH is coercive and weakly lower semi-continuous on C1 with respect to H I ,2(B; JR.3). By Theorem 1.3.3 f3H is attained at X E C1. We show that X for all IHIR < 1 is a relative minimizer of DH with respect to the topology of H I ,2 n L oo (B;JR.3). Let
tp E H~,2 n L oo (B;JR.3) satisfy 0::; tp::; 1. Then for
f
E [0,1]
Xf=X-ftpXECH,
and by minimality of DH(X) and (3.4) we obtain 0::;
~DH(Xf)bo = -
JVX
V(tpX)
+ 2HX,,/\ Xv· tpXdw
B
JV C~12) Vtp + [IVXI 2+ : ; -J C~12)
=-
2H Xu /\ Xv· Xl tp dw
B
Vtpdw.
V
B
I.e. the function equation
IXI 2 E
H I ,2 n Loo(B;JR.3) is a weak sub-solution to the Poisson
::; o. R2) + = sup (IXI 2- R2, 0) _~(IXI2)
Choosing tp =
(IXI 2 -
JIv (lX12 -
E H~,2 n L OO we obtain that
R2)+12 dw
= 0,
B
I.e. IX I ::; R a.e. on B, and (3.2) holds. It follows that X lies interior to CH with respect to the topology of H I ,2 n Loo(B; JR.3) and is a relative minimizer of DH in this class. In particular, by Remark 2.4. ii) X solves (1.1)-(1.3).
=
The case IHIR 1 : Choose a sequence {Hm} of numbers Hm-+ H, IHmlR < 1, and let & be a corresponding sequence of Hm-surfaces characterized by (3.2), (3.3). In particular, by (3.2) Xm E C1. By Lemmata 3.2, 3.3 therefore we may assume that
and by (3.3), letting X E CH C CHm be a minimizer of D H :
m-+ 00
::;
lim inf DH.,.(X) m-+oo
m-+ 00
= DH(X) = f3H.
103
III. The existence of surfaces of prescribed constant mean curvature
Hence
i.e.
X minimizes DH in CH, and in particular
X is conformal, cpo Remark 2.4 and Lemma 1.2.4.
o = (dDHm (Xm)' I{» = (dD(Xm), I{» + 2Hm (dV(Xm), I{» ->(dD(X), I{» + 2H(dV(X), I{» = (dDH(X), I{» as m -> 00, by weak continuity of dD, dV : H 1 ,2(Bj JR3) -> (H~,2(Bj JR3))· and uniform boundedness l(dV(L,.), I{»I ::; c D(Xm)D(I{>)1/2 ::; c. I.e. dDH(X) o E (H~,2(BjJR3)r, and X weakly solves (1.1)-(1.3).
=
o The next existence result is due to Steffen [3] , improving an earlier result of Wente
[1] . Theorem 3.4: Let r be a Jordan curve in some Xo E C(r) there holds
JR3, H E JR. Suppose that for
(3.6) Then there exists a solution X. to (1.1) -(1.3) which is a relative minimizer of DH in the class {X E C(r)ID(X) ::; 5D(Xo)}. The proof of Theorem 3.4 will be given in a later section, cpo Theorem IV.4.13. Note that the constant 2/3 in (3.6) is not optimal. In fact, Remark IV.4.14 shows that the assertion of the theorem remains true provided H2 D(Xo) < 27r /3 + € for . some € > 0 depending on r. It is conjectured that it suffices to assume that H2 D(Xo) < 7r, which is optimal in the case of a circle.
104
B. Surfaces of prescribed constant mean curvature
4. Heinz' non-existence result. The necessity of the smallness condition in the existence theorems of the preceding section is illustrated by the following result of Heinz [2]: Theorem 4.1: Let f C IR? be a rectificiable Jordan curve of length L(r), and suppose that for some Xo E C(f) and some unit vector no E IR? the number
Co
=
J
XOu /I. Xo.v . nodw
> o.
B
Then if HE JR satisfies
JHJ > (L(f)j2co )
there is no solution to (1.1)-(1.3).
=
Remark 4.2: In particular, if f BB is the unit circle in JR2 we may let Xo(w) == w, no (0,0,1). Then L(f) 21t', Co 1t', and it follows that f cannot span H -surfaces with curvature JHJ > 1.
=
=
=
Proof: We present the proof in case f E cl,a. Then any H -surfaces X E C(f) is of class Cl(B; JR3), cpo Theorem 5.5, and from (1.1)-(1.3) we obtain
2H
J
J
B
B
Xu /I. X" . nodw =
:; J
b"X . nodw =
J
Bn X · nodo
U
JBnXJdo =
lJB
JI:>xl
do = L(f).
lJB
By invariance of the integral
Co
= no·
J
Xou /I. Xo"dw
B
under orientation preserving changes of parametrization we may assume that X - Xo E H;;,2(B; JR3). But then by (1.9)
J =J
(Xu /I. X" - Xou /I. X o,,) . nodw =
2
B
[(X - Xo)u
/I.
(X + X o)" + (X + Xo)u
/I.
(X - Xo),,] . nodw = 0
B
and it follows that 2co JHJ::; L(f).
o
Remark: It is not known whether a similar result holds for the Dirichlet problem (IV.l.l)-(IV.1.2) below.
105
lIT. The existence of surfaces of prescribed constant mean curvature
5. Regularity. Regularity of weak solutions to (1.1) - (1.3) with minor modifications can be obtained as in the case of minimal surfaces (H 0) once the following result due to the Wente [5, Lemma 3.1) has been established.
=
Theorem 5.1: Let
(5.1) Then
D-Z Z is continuous on
=
B.
Any weak solution
Corollary 5.2:
in B.
X E C(r) to (1.1)-(1.3) is continuous on
B. Proof: Decompose X = Xo + Z, where Xo E Co(r), i.e. D-Xo = 0, and Z E H;;,2(Bj JR3). By the maximum principle and (1.3) Xo is continuous. Now Z weakly solves D-Z 2H Xu /\ Xv in B
=
and therefore is continuous by Theorem 5.1 .
o The proof of Theorem 5.1 will be a consequence of the following Under the assumptions of Theorem 5.1 we have
Lemma 5.3: and
Z E LOO(Bj JR3)
Proof: First note that (5.1) is conformally invariant. Indeed, if 9 is a conformal diffeomorphism of B and if Z' Z 0 g,
=
e
Jv J(
ZVedw
B
B
where
=
J
= v Z'Ve' dw, B
1/Jv
=
+ 1/Ju /\
J(
/\
1/J~ + 1/J~ /\
B
e =e
0
g.
Since for any wEB there is a conformal diffeomorphism of B mapping w into o it hence suffices to prove the estimate
(5.2)
IZ(O)I :::; 2 (D(
for solutions Z of (5.1) with a Lebesgue point at
0, i.e. such that
106
B. Surfaces of prescribed constant mean curvature
(5.3)
!
lim r- 2
r--+O
IZ(w) - Z(O)ldw = O.
Br(O)
Introducing polar coordinates (r,¢) on
B, note that e.g.
CPu =cos¢ CPr - (l/r) sin ¢ CP.p,
CPv =sin¢ CPr
+ (l/r)cos¢ CP.p,
whence (5.1) may be rewritten in the form:
For
f
>
0 and some unit vector a E
J:If
let
Approximating ~ in H;'2 n Co by smooth functions we can justify inserting as a testing function in (5.1). This gives, cpo (1.9):
~
!
l/f
Z· a d¢
!
=
oBf(O)
'V Z'V~ dw
B 1 211'
!! = - ! ! (~r =!
(CPr I\"p.p +"pr 1\ cp.p) . ~d¢ dr
=-
o
0 1 211'
o
0
I\"p.p +"pr 1\ ~.p) . cpd¢ dr
1 211'
!(1/r)al\"p.p.cpd¢dr.
f
0
For r E [f, 1] let
~(r) = 1/(21rr)
!
cpd¢.
oBr(O)
Then for a.e.
r E
!
[f, 1] by Holder's and Poincare's inequalities:
211'
o
!
211'
a 1\ "p.p . cpd¢
=
0
a 1\ "p.p . (cp -
~)d¢
107
III. The existence of surfaces of prescribed constant mean curvature
Hence
1 2,..
( ::; 211"
ff
1 2,..
IV1/I1 2rd4> dr .
e 0
ff
) 1/2 IVcpl2r d4> dr
e 0
::; 411" (D(1/I)D(cp))1/2 . But by (5.3)
211"Z(0) - (lie)
f
Z d4> ::; (lie)
8Be(O)
if we let e
-+
f
IZ - Z(O)I d4>
-+
0
8Be(O)
0 suitably, and (5.2) and the lemma follow.
o Remark 5.4:
The optimal constant in Lemma 5.3 is
If cp and 1/1 are conformal (e.g. if cp = 1/1 = X where one even finds the estimate
X solves (1.1) - (1.3))
IIZIILOO ::; (1/211") (D(cp)D(1/I)) 1/2 , which is best possible, cpo Wente [5]. Proof of Theorem 5.1:
Introducing tangential differences
Z(h)(r,4» == Z(r,4> + h) - Z(r,4», cp+(r,¢) == cp(r,¢ + h) we have
.6.Z(h) (5.4)
+ 1/Iih) 1\ cp~h) + cp~h) 1\ 1/1" + 1/Iu 1\ cp~h) + CPu 1\ 1/I£h) + 1/Iih) 1\ CP'l)'
=cp~h) 1\ 1/I~h)
By Lemma 5.3
and Z is uniformly continuous in tangential direction. Given e> 0 choose 61 > 0 such that for any r E [0,1] IZ(r,
E
108
B. Surfaces of prescribed constant mean curvature
Note that for any connected interval
I01
oflength 61
1
ess inf
I01
1
J
JJ
IV'ZI2rdrl4>=4>o :::; 61'1
IV'Zl2rdr d
[01
0
0
62 be such that
)46 2 61'1 D(Z) < f. Then for any pair of points distance I
(r,
r'l < 62 ,
IZ(r,
:::;2f +
J
IV'Z(r, <po)ldr
r
Taking the infimum with respect to
I.e.
Z is continuous on the closed annulus
{wit:::; Iwl:::;
I}.
Finally, consider Z in Cartesian coordinates again, and for an arbitrary unit vector e E JR2 let
Z(h)(W)
= (Z(w + he) -
Z(w)).
i
Note that for Ihl < Z(h) is defined on the ball B%(O) and satisfies a system like (5.4) with boundary values uniformly tending to 0:
IIZ(h) II
L
OO
(lW.1(O))
--+
0 (h
--+
0).
•
zi
h) + Z~h) with L:!.Zih) = 0 and Z~h) E H~,2(Bt(0)) we Decomposing Z(h) = conclude from Lemma 5.3 and the maximum principle that IIZ(h) IIL OO (B.1 (0»
•
--+
0 (h
--+
0) .
This concludes the proof.
o
III. The existence of surfaces of prescribed constant mean curvature
109
With the aid of Theorem 5.1 the proof of Theorem 1.5.1 conveys to and we obtain the following result, cf. Hildebrandt [3, Satz 7.1 ]:
H -surfaces
Theorem 5.5: Let f C 1R? be a Jordan curve of class C'TTt,O, m ~ 2, 0 < < 1, and suppose X E C(r) weakly solves (1.1)-(1.3) for some H E JR, then X E Cm,O(B; JR3).
a:
In general, we do not obtain a-priori bounds. However, for "small" solutions the following variant due to Heinz [3] of the isoperimetric inequality Theorem 1.4.9 in conjunction with Lemma 1.4.3 allows to formulate a-priori estimates for normalized H -surfaces X E C*(f) analogous to Theorem 1.5.1. Theorem 5.6: Let f C BR(O) C JR3 be a rectificiable Jordan curve of length L(f), and let H E JR satisfy IHIR < 1. Then for any H -surface X E C(r) with IIXIILoo :5 there holds the estimate
rkT
47rD(X)
:5
1 + IHiR 2 1 _IHIRL(r) .
We prove the qualitative estimate for smooth f. Proof: Note that as in the proof of Theorem 3.1, by the maximum principle for sub-harmonic functions there holds IIXIILoo:5 R for any H -surface X E
C(r) with
IIXIIL oo :5
rkT.
But then by (3.4)
+ 2HX.,. /\ Xv], Xdw
0= ![-b.X B
!
!
B
U
= I\7XI 2 + 2HX.,./\ Xv' Xdw ~2(1 -IHIR)D(X) - IIXllL oo
8nX· X do
! 18
nXldo.
aB
Since by (1.2) and Theorem 5.5
! 18
nXIdo =
aB
! I:f/I xl
do = L(f)
aB
the claim follows.
o Although for H -surfaces in general no a-priori bounds on the area or their modulus exist, some partial results are known. The next result is due to Wente [5]:
110
B. Surfaces of prescribed constant mean curvature
Theorem 5.7: Suppose r C BR(O) C JR3 is a Jordan curve, X E C(r) an H -surface spanning r. Then
JJXJJL"" ::; R + (1/27r)JHJD(X). The bound cannot be improved.
For embedded
H -surfaces Serrin [1] states the following:
Theorem 5.8: Suppose X E C(r) is an embedded Jordan curve r C BR(O) c mJ. Then
H -surface spanning a
JJXJJL"" ::; R + 2/JHJ. The bound cannot be improved. Serrin's proof is based on the Alexandrov reflection principle and a-priori estimates for the non-parametric mean curvature equation (describing H -surfaces which can be represented as graphs over JR2). However, there appear to be gaps in his argument which still wait to be filled. (The surface X might wind around and "reenter" through the curve r in such a way that upon reflecting a portion of X in a plane this reflected surface might touch X with the "wrong" orientation for the maximum principle to be applicable. This is impossible for closed surfaces. ) Remark that from the surfaces discovered by Wente [6] one can construct immersed surfaces of genus 1 having constant mean curvature but which do not satisfy the bound of Theorem 5.8.
IV. Unstable H-surfaces 1. H - extensions. In the analysis of unstable minimal surfaces we relied on the existence of harmonic extensions of admissible parametrizations of r in order to reformulate the Plateau problem in terms of a variational problem on a convex set. To imitate this procedure for H -surfaces we now consider Dirichlet's problem for the H -surface system:
~X=2HXuI\Xf1
(1.1) (1.2)
in B,
X =Xo on 8B
or , equivalently, find
X E Xo
+ HJ,2(Bj JR3)
such that
Recall that for H = 0 the harmonic extension X of Xo E H 1 ,2 n L OO (BjJR3) is uniquely characterized by the relations
X E X0
(1.3)
_
(1.4)
D(X)
+ H 1 ,2(B· JR3) 0
,
,
= inf{D(X)IX E Xo + H~,2(BjJR3)}.
Moreover, X depends differentiably on Xo in H 1 ,2 n L OO by linearity of the Poisson equation, the variational characterization (1.4), and the maximum principle.
Due to the nonlinear character of the H -surface system (1.1) for Ht-O certain smallness conditions will have to be satified. Under hypotheses similar to those of Theorem III.3.1, however, we can establish the existence of H - extension operators with properties analogous to the harmonic extension operator. Moreover, Hextensions can be characterized by a variational principle consistent with (1.3)-(1.4).
The following result again is due to Hildebrandt [2]: Theorem 1.1: Let Xo E H 1 ,2nL OO (BjJR 3), HE JR. Let R = IIXoIILoo, and suppose that IHIR ~ 1. Then there exists a solution XH of (1.1), (1.2) characterized by the conditions that
(1.5)
IIXHllLoo ~ R,
(1.6)
DH(XH)
= inf {DH(X) I X
E Xo
+ H;,2(Bj JR3),
IIXllL oo
~
I!I}'
112
B. Surfaces of prescribed constant mean curvature.
If
Xo
IHIR < 1 then
XH is a relative minimizer of
+ H;·2 n L OO (B;JR3).
DH on the affine space
Proof: By Lemmata 111.3.2, 111.3.3 DH is coercive and weakly lower semicontinuous (with respect to the H 1 •2 -topology) on the space SH:= {X E Xo + H;·2(B; JR3)IIIXIlLOO ::; l/IHI}. Hence DH achieves its infimum at X H E SH.
If IHIR < 1 as in the proof of Theorem 111.3.1 IIXHII::; Rand XH yields a relative minimum of DH on Xo + H;·2 n LOO(B; JR3). In particular,
dDH(XH) whence by density of XH solves (1.1), (1.2). The case
IHIR
=1
=0
C:;"(B;JR3)
E (C:;"(B;JR 3»)',
H;·2(B;JR3)
in
and Remark III.2.4.ii)
is treated as in Theorem 111.3.1.
o In order to establish continuous dependance of the solution XH of (1.5), (1.6) on the data Xo we now investigate the behavior of DH near a relative minimizer in Xo + H;.2 n L OO (B;JR3). The following result is due to Brezis-Coron [1] : Lemma 1.2: If X E Xo +H;·2(B;JR3) is a relative minimizer of DH with respect to H;·2 n L OO (B;JR3) then X is a strict relative minimizer of DH on Xo + H;.2(B; JR3) and for some 8> 0 there holds
d 2DH(X)(rp, rp) ~ 8D(rp), Vrp E H;·2(B; JR3). Proof:
By density of C,:, in H;·2 clearly
d2DH(X)(rp, rp) ~ 0 for all rp E H;·2(B; JR3).
(1.7) Let 8
= inf {d2DH(X)(rp, rp)1 rp E H;·2(B;JR3),
D(rp)
= I} ~ o.
The lemma follows once we establish that 8 > o. Suppose by contradiction that 8 = o. Let rpm E H;·2(B;JR3) be a sequence such that D(rpm) = 1 while
d2DH(X)(rpm, rpm)
--+
We may assume that rpm!£. rp weakly in 111.2.3 and Example 1.3.4
0::; d2DH(X)(rp, rp)
m
--+ 00
m --+
00
0 (m --+
00).
H;·2(B; JR3) whence by Theorem
= 2D(rp) + 2H d2V(X)(rp, rp)
113
IV. Unstable H-surfaces.
Since by Theorem 111.2.3
it follows that D(
-+
D(
-+
(1.8) By (1.7)
i.e.
b.
+
in B.
From Lemma 111.5.2 we infer that
DH(X) ~ DH(X + t
= DH(X) + 2HtlV(
I.e. and all functions X +t
b.(X + t
= 2H(X + t
In
B.
Differentiating twice with respect to t we obtain that
d 2D H (X)(
> O.
o Corollary 1.3: If Xl, X 2 E Xo + H;,2(Bj JR.3) are relative minimizers of DH on X o +H;,2(BjJR.3), with respect to variations in H;,2n£OO(BjJR.3), then Xl
= X 2•
Proof:
Let
DH(XI) =DH(X2 +
114
B. Surfaces of prescribed constant mean curvature.
we obtain that
by Lemma 1.2. Hence
I{)
= O,X1 = X2. D
Corollary 1.4: Let H, R E JR satisfy IHIR < 1. There is a unique differentiable and bounded map 'T/H of the space {Xo E H1.2 n LOO(Bj JR3) I IIXollLoo :::; R} into itself which associates to any Xo the unique solution XH to (1.1), (1.2) characterized by (1.5), (1.6). Proof: H~·2(Bj
Existence and uniqueness of X H = 'T/H(Xo) = Xo JR3) follows from Theorem 1.1, resp. Corollary 1.3.
+ Zo,
By Lemma 1.2 and the implicit function theorem, for any such XH exists a unique local solution Z(X) of the equation
dDH(X for
Zo
= Zo(Xo) E
= Xo+Zo
there
+ Z(X)) = 0 E (H1.2(Bj JR3)) * .
X in an H 1 •2 n LOO(Bj~)-neighborhood of Xo.
f(X, Z)
= dDH(X + Z)*
E H;·2(Bj JR3)
is analytic in a neighborhood of (Xo, Zo), and the estimate
Iz
following from Lemma 1.2 shows that f(X o, Zo) is an isomorphism of H;·2(Bj ~). Letting 77H(X) X + Z(X), by Lemma 1.2 and continuous dependance of ~ DH(X) on X E H 1•2(Bj ~), 77H(X) is a relative minimizer of DH on X + H~·2(Bj~). Hence by Corollary 1.3 77H 'T/H, and 'T/H is a smooth map from H 1•2 nLOO(Bj~) into H 1•2(BjJR3). Smoothness of T/H into L OO (BjJR3) follows from Lemma 111.5.3:
=
=
b.Zo = 2HXHu
1\
X H., in B,
and by Lemma 111.5.3 the operator b. -1 (with homogeneous Dirichlet data) composed with the map X 1-+ Xu 1\ X., is smooth from Hl.2(B;.JR3) into LOO(B;~).
115
IV. Unstable H-surfaces.
Finally, boundedness of TJH is immediate fron (1.5), (111.3.5).
o Remark 1.5: i) Analogous to Theorem 111.3.4 one also obtains relative minima of DH on Xo + H;,2(BjlR?) provided the condition (1.7) is satisfied for H, cpo Steffen [3] . Since (1.7), however, cannot hold simultaneously for all X E Co(r) for any rectificiable Jordan curve in JR3, this condition is not suited for constructing H -extensions on the whole class Co(r). By a result of Jager [1] actually for IHIR < 1 the solution XH the only solution to (1.1), (1.2) with IIXHlloo::; R.
ii)
= TJH(Xo )
is
116
B. Surfaces of prescribed constant mean curvature.
2. Ljusternik - Schnirelman and Morse theory for "small" H-surfaces. In analogy with Section 11.2, 11.4 we now develop a critical point theory for "small" solutions to the Plateau problem for surfaces of prescribed mean curvature H, i.e. for solutions X to (111.1.1) -(111.1.3) spanning a Jordan curve r E BR(O) C I£t3 that satisfy the extra condition IIX IILoo ~ R. Recall the definitions of the sets M (resp. Mt) of monotone reparametrizations of [0,21r]=8B (normalized with respect to the conformal group action). For brevity, again denote T:= Hl/2 ,2 n C O (IR/21r) with norm I· I . Let 'Y E C r , r ~ 2, be a reference parametrization of r and let Xo : M
-+
Co(r)
= {X E c(r) I~X = O}
be the map :e
1-+
= h(-y
Xo(:e)
0
:e),
where h denotes harmonic extension. By Lemma 11.2.5 and the maximum principle the map Xo extends to a map of class C r - l of {id} + T ipto
Now suppose that
IHIR < 1. Composing the map Xo with the Corollary 1.4 we obtain a map XH: M
:e
H -extension map
-+
C(r)
1-+
XH(:e)
'T/H
constructed in
='T/H(Xo(:e»
whose image lies in the set
Moreover, we may let
Lemma 2.1: If 'Y E cr, r ~ 2, XH extends to a map of the affine space {id}+T into H 1 ,2 n L OO (BjIR 3 ) of class cr- l . Proof:
By smoothness of 'T/H XH inherits the properties of Xo.
D Lemma 2.2: If 'Y E cr, r ~ 2, the functional EH extends to a functional on {id} + T with derivative at XH = XH{:e) for :e E M by (dEH(:e),{)
=
f
8B
8.. XH'
d~'Y(:e)·{do
cr- l . given
117
IV. Unstable H-surfaces.
for all
eE T.
Subsets of Mt where EH is uniformly bounded are bounded in H 1 / 2 ,2 and relatively compact with respect to the CO-topology. Proof: The last statement follows from Lemma 111.3.2, Lemma 11.2.8, and Remark 11.2.4. Moreover, by Lemma 2.1 and analyticity of DH the assertion that EH E C .. - 1 is obvious.
Compute for smooth :c, using the fact that dXH(:c)· e = dXo(:C) . e = ~-y(:c) . e on 8B:
XH
= Xo
on
8B implies that
(dEH(:C)'e) =(dDH(XH), dXH(:C) ·e) =
=/
VXH V(dXH (:c) . e)dw
+
(2H/3) / XHu 1\ X H" . (dXH(:C)' e) +
B
B
+ (XHu 1\ (dXH(:C)' e)" + (dXH(:C)' e)u 1\ XHv) . XH dw = / [-~X + 2HXHu 1\ X H,,]· (dXH(:c)· e) dw B
+/
d~-y(:c)·e do
8n X·
{}B
+ (2H/3)
/ (v. XHu - UXHv) 1\
(d~ -y(:c) . e) . X H do.
(}B
=
Now the first and the last term on the right vanish since by definition ~XH 2HXHu 1\ X H" In B, while V· XHu - UXH" -~XH is co-linear with
=
~-y(:c) along 8B. The general case follows by density of smooth monotone maps in M . Note that all integrals exist in the distribution sense. In particular, since XH solves (111.1.1) the estimate for harmonic cp E H 1,2 n LOO(B; J1i3)
/ 8n XH' cp do (}B
=/
VXHVcpdw
+ 2H /
B
XHu 1\ XHv . cpdw
B
+ D(XH)IHlllcpIiLoo ~ CD(XH )1/2 Icp1H1/2 ,2({}B) + D(XH) IHI IIcpliLoo ~ cD(XH )1/2 D(cp)1/2
shows that
8n XH exists as a distribution on H 1/ 2 ,2 n L OO (8B; J1i3).
o Define gH(:C)
=
sup (dEH(:C)' :c - y).
"EMt 1_-,,1<1
118
B. Surfaces of prescribed constant mean curvature.
Analogous to Proposition 11.2.9 one can show: Lemma 2.3:
Z
E Mt satisfies gH(Z)
=0
iff X
= XH(z)
solves (III.l.l)-
(III. 1.3). EH satisfies the Palais-Smale condition on Mt:
Moreover,
Lemma 2.4: while gH(Zm)
By Lemma 2.2 we may assume that for some
H 1/2,2
Xmo
and uniformly, whence also
-+
IEH(zm)l ~ c uniformly
Any sequence {zm} C Mt such that 0 (m -+ 00) is relatively compact.
-+
Z
Xm ~ X
E Mt
~
Zm
weakly in
Z
weakly in
H I ,2(B; JR3)
and
Xo uniformly on B. As in the proof of Lemma 11.2.12 Xmo - Xo
where Im
-+
= d
0 strongly in H 1 / 2
,2
z)
+ Im
on BB ,
n GO(BB).
Hence
0(1)
~ gH(zm)lzm -
zi
~
BnXm' !"Y(zm)(zm - z) do
OB
J =J =J =
J
BnXm . (Xmo - Xo) do + 0(1)
OB
V Xm V(Xmo - Xo) dw + 2H
B
J
Xmu 1\ X m" . (Xmo - Xo) dw
+ 0(1)
B
V (Xm - X) V (Xmo - Xo) do + 0(1).
B
By orthogonality
J
V XV Z dw
= 0,
V X E eo(r), Z E H;,2(B; mJ)
B
the latter implies that
J
IV(Xmo - XoWdw
~ 0(1).
B
I.e.
X_o
-+
Xo in H I ,2(B; mJ), and
Z_ -+ Z
in H l /2,2 n GO as claimed. []
119
IV. Unstable H-surfaces.
Applying Theorem II.1.13 we hence obtain the following result which strengthens results of Heinz [5] and Strohmer [1] on the existence of " small" unstable H -surfaces in CH(r) : Theorem 2.5: Let r C BR(O) c JR3 be a Jordan curve of class e 2 , HE lR, IHIR < 1. Assume DH admits two distinct relative minima Xl. X 2 E CH(r). Then either there exists an unstable solution X3 E CH(r) to the Plateau problem (III.1.I) - (III.1.3) or DH(X 1 ) = DH(X2) = {3 and X lI X 2 can be connected in any neighborhood of the set of relative minima X of D H in CH(r) with DH(X) = {3.
In order to extend the Morse theory of Section 11.4 to "small" solutions to the Plateau problem for surfaces of constant mean curvature note the following Lemma 2.6: Suppose r C BR(O) C JR3 is of class parametrization ,,(, and H E lR satisfies IHIR < 1.
e5,
with reference
Then the functional EH E e 3 ( { id} + T), and at any critical point (where gH(:Z:o) = 0) the expansions hold:
:Z:o E Mt
= EH(:Z:o) + 1/2 d2EH(:Z:o)(:Z: - :Z:0,:Z: - :Z:o) + 0(1:z: - :Z:01~/2 ), (dEH(:Z:)':Z: - y) = d2EH(:Z:o)(:Z: - :Z:0,:Z: - :Z:o) + 0(1:z: - :Z:01~/2 ), EH(:Z:)
for all :z:, y E Mt, such that
I:z: - Yl1/2 ~ I:z: - :z:011/2
The proof is the same as that of Lemma 11.4.2. No,! let
Tt ~ {eE Tile. , do ~ 0, V, E T;'G} Ht
= H 1 / 2 ,2 _
clos(Tt)
as in Section 11.4 and consider
Lemma 2.7: Under the hypotheses of Lemma 2.6 if :Z:o E Mt is critical, the bilinear form d 2EH(:Z:o) extends to a bilinear form on Ht and induces a
decomposition
H! ffi H! C
Ht
= Hl ffi H! ffi
H!. Moreover, dim (H! ffi H!) <
00
and
e l (lR/21f).
The proof of Proposition 11.5.6 conveys with minor modifications.
As before a critical point :Z:o E Mt will be called non-degenerate if d2EH(:Z:o) induces an isomorphism of Ht.
120 Lemma 2.8:
B. Surfaces of prescribed constant mean curvature.
Under the hypotheses of Lemma 2.6, if
degenerate critical point of decomposition of Ht at
EH :Co,
:Co
E Mt
is a non-
=
and Ht Hi ffi H! denotes the standard then there exists a neighborhood 1L of 0 in
H! such that
=
Proof: :Co induces a regular solution X XH(:C o ) of (111.1.1)-(111.1.3). Since branch points of X give rise to forced Jacobi fields E Ht with
e
(cp. Sollner [1]) our non-degeneracy assumption implies that X is immersed over B. In particular, ~ c > 0, and the claim follows from Lemma 2.7. o
-ip:c
o By Lemma 2.6 and Lemma 2.8 Theorem 11.3.6 applies to obtain the following result: Theorem 2.9: Suppose f C a diffeomorphism 'Y E C 5 , and that all small H-surfaces X non-degenerate critical points (11.3.4) hold.
EH on
Mt and we
BR(O) C JJf is a Jordan curve parametrized by HEIR satisfies the bound IHIR < 1. Suppose XH(:c) E CH(r) bounded by f correspond to of EH in Mt. Then the Morse inequalities
= :c
Remark 2.10: As in the case of minimal surfaces the non-degeneracy of a parametrization of an H -surface X XH(:C) E CH(f) as a critical point of EH will be equivalent to the non-degeneracy of X on the space dXH(:c)(Ht) of surfaces "tangent" to CH(f) at X and the Morse indeces correspond, cpo Remark 11.4.9.
:c
=
121
IV. Unstable H-aurfacea.
s. Large solutions to the Dirichlet problem. By Corollary 1.3 solutions to (1.1)-(1.2) of minimum type are unique. However, any time there exists a relative minimumof DH on X o +H;,2(Bj,m3) for H",O the global behavior of DH as a cubic functional lets us expect a further solution which is not of minimum type. The following result is due to Brezis-Coron [1] and the author [3] with an extension by Steffen [4]. More precisely, in Struwe [3] the thesis was obtained for" admissible" Xo E H 1 ,2 n L OO (Bj,m3) and small H",O, while Steffen was able to show that all non-constant Xo are "admissible" in that sense. Independently and almost simultaneously Brezis and Coron found non-uniqueness for non-constant Xo E H1,2 n L OO (BjIR?) and H E JR satisfying the bounds < IHIR < 1, where R = IIXo IlL 00. Moreover, their proof extends to the general case considered below.
°
Theorem S.l: Let Xo E H 1 ,2 n LOO(Bj ,m3), H E JR, and suppose that =t= const, H",O, and that DH admits a relative minimum XH on Xo + H;,2(Bj ,m3). Then there also exists an unstable solution XH of (1.1)-(1.2) on Xo + H;,2(Bj,m3).
Xo
°
Theorem 3.1 in particular applies if < IHI IIXollLoo < 1 or H2 D(Xo) < 27r/3. The result is optimal in the sense that if Xo == const or H = the solution to (1.1)-(1.2) is unique. This is a consequence of a result due to Wente [4] :
°
Theorem S.2: Suppose X E H;,2(Bj,m3) satisfies (1.1)-(1.2) for some HE JR with boundary data Xo == 0. Then X == 0. Proof:
Extending X by reflection
Iwl:::; 1 j;p"' Iwl> 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 <) having the properties i), ii), iii) of Lemma I1.1.9. Suppose by contradiction that Kfj consists of relative minima of E only. Then Kf3 is relatively open (and trivially closed, by continuity) in
Let
N be a neighborhood of Kfj such that
for any that
f
> 0, and let
f, <)
N
~ Z1
and
be chosen correspondingly. Select a path PEP such supE(z) <{3+f. II:Ep
Then
p = <)(p, 1) E P
by property i) of Lemma II. 1.9, while by property iii) p' C N U M f3 - f
•
Nand Mfj_f being disjoint, it follows that either p' C N or p' C Mfj_f. But E p', and since Z1 ¢ N the first case cannot occur, while the second contradicts the definition of {3. The contradiction proves the theorem. Z1
o We now apply Theorem 3.3 with E = DH on T = Xo + H;,2(Bj JK3), it = XH. The following result is essentially due to Wente [3]. In the generality stated below it was proved by Brezis-Coron [2]:
123
IV. Unstable H-aurfaces.
Lemma 3.4: For any H=I=O the functional DH admits a surface Xl such that DH(X l ) < DH(XH). Moreover, letting P and f3 be given by (3.1), (3.2) , if Xo ~ const. we have:
f3 <
DH(XH )
Note that 411"/(3H2) is the "energy" conformal representation.
+ 411"/(3H 2). DH
of a sphere of radius
l/IHI
In
The proof below is essentially due to Brezis-Coron [2] : Proof: If Xo ~ const clearly X H ~ const and at some point Wo E B we have V XH (wo)=I=O. By translation and rotation of coordinates we may assume that Wo 0, and
=
:UXH(O)
= (a l ,a2 ,a3 ),
:vXH(O)
= (bl,b 2,b3 )
satisfy the condition that (3.3)
For
>0
f
now let rpf(U,V)
=
2 f
2~
+'1.1. +v
2
(U,V,f)
be a conformal representation of a sphere of radius by stereographic projection from the "north pole".
1 around
(0,0,1), obtained
Also let ~ E C:;"(B) be a symmetric cut-off function such that ~(w) == 1 near W O.
=
~
= ~(-w)
and
Consider the family
X: For
f
= 0,
X:
= X H + t~rpf E Xo + H;,2(Bj IR?).
can be pictured as a sphere of radius
t
attached to
XH
at
XH(O).
Compute, using (111.2.13) DH(X;)
= DH(XH ) + DH(t~rpf) + t 2Hd2V(XH )(~rpf, ~rpf)
=DH(XH)
+ t 2D(~rpf) + 2t 3 H V(~rpf) +
2t2 H
f
XH .
(~rpf).. 1\ (~rpf)" dw.
Vrpf
+ Irpfl2lV~12dw
B
Now
D(~rpf) = D(rpf) + 1/2
J(e -
1)IVrp f 12
+ 2~ V~ rpf
B
~ D(rpfj m2) V(~rpf) = V(rpf)
+ 0(f2) = 411" + 0(f2),
+ 0(f3)
= V(rpfj m2) - 1/3
J rp~
.R2 \B
1\ rp! . rpf dw
+
0(f3)
124
B. Surfaces of prescribed constant mean curvature.
Expanding XH(U,tI)
= XH(O)+au+btl + 0(r 2 ),
where r2=u 2 +tl 2
:
=
2H / XH . (e
= 2H / (XH(O)
+ au + btl) . (e
(e
B
+ 2H /
0(r2)(e
B
By (III.1.9) the first term
=H /
(a" (e
B
which by antisymmetry
e.
The last integral vanishes by symmetry of f > 0 we can estimate
/ (f2 :3r2 )2e dW 2: / B
(f2
Moreover, for sufficiently small
~3r2)2edW
BdO)
2:clf > with a uniform constant
Cl
0
> o. On the other hand since
may estimate
/ 0(r2) . (e
s
B
s
! ! !
0(r2) . IV
+ 0(f2)
B
:::; C
f2r2 (f2
+ r 2 )2 dw +
B
:::; C
dw
BdO) :::; C2.
f21lnfl
+
C
0(f2)
! ;:
B\Bf(O)
+ 0(f2).
dw + O( f2)
125
IV. Unstable H-surfaces.
Hence for sufficiently small
E
>0
DH(xf) ::; DH(XH) + (411" + 4H(a l
+ b2)ClE + c2E21ln EI + 0(E2» t 2
+ 2H (411"/3 + 0(E3» t 3 •
=
Clearly if Ht -+ -00 we have DH(Xn -+ -00 . Hence if Hf.O a surface Xl exists as claimed. Moreover, suppose H < O. Then, if Xo f. const. and l if (3.3) holds, for sufficiently small E > 0 the value sup DH(Xn is achieved at
xt
t~O
to ::; l/IHI- C3E, where C3 > O. Hence in this case sup DH (X[) t~O
< DH(XH) + 411" /(3H 2),
and the lemma follows. The case H
>0
may be treated similarly.
D Finally, we establish the local Palais-Smale condition
(P.S.)f3.
Lemma 3.5: Let H f.0 and suppose that X H is a relative minimizer of DH on X o +H,!,2(B; 1R?). Then any sequence {Xm} in X o +H,!,2(B; JR3) such that
is relatively compact. Proof: To show boundedness of there exists 6 > 0 such that
Now let
Ipm
= Xm -
{Xm}
in H 1 ,2 observe that by Lemma 1.2
X H and expand
DH(Xm ) = DH(XH + Ipm) = DH(XH ) + 1/2 d 2 DH(XH)(lpm, Ipm) + 2H V(lpm), (dDH(Xm ), Ipm) =d2DH(XH)(lpm, Ipm) + 6H V(lpm) = 0(1) D(lpm)1/2. Subtracting three times the first line from the second there results
3 (DH(Xm) - DH(XH» - 0(1) D(lpm) and
1/2
= 1/2 d 2 DH(XH )(lpm, Ipm) ~ 2"1 6D(lpm)
D(lpm)::; c uniformly.
Xm ~ X weakly in H 1,2(B; JR3). By weak continuity of D H , cpo Theorem 111.2.3, dDH(X) = 0, whence the cubic character of DH guarantees that Hence we may assume that
(3.4)
126
B. Surfaces of prescribed constant mean curvature.
= Xm -X
Now let 'l/Jm
~ 0 weakly in H;,2(Bj JR3). Expanding, using (III.2.13)
and Theorem 111.2.3, we obtain:
DH(Xm) = DH(X) + DH('l/Jm) + H d2V(X)('l/Jm,'l/Jm) (3.5) = DH(X) + DH('l/Jm) + 0(1), 0(1) (dDH(Xm),'l/Jm) (dDH('l/Jm),'l/Jm) + 2Hd2V(X)('l/Jm,'l/Jm) (3.6) = 2D('l/Jm) + 6HV('l/Jm) + 0(1),
=
where 0(1)
-+ 0
=
(m -+ 00).
In particular, for m
~
mo by (3.4-5) : 4'11"
DH('l/Jm) :5 DH(Xm) - DH(X) + 0(1) :5 c < 3H2' while from (3.6) we deduce that
3DH('l/Jm)
= 3D('l/Jm) + 6H V('l/Jm) = D('l/Jm) + 0(1).
I.e. for (3.7) But now (3.6) again and the isoperimetric inequality Theorem 111.2.1 imply that
2D( 'l/Jm)
(1 -
H2 ~~'l/Jm») :5 2D( 'l/Jm)
In view of (3.7) this implies that D('l/Jm)
-+
+ 6H V( 'l/Jm) = 0(1).
0 (m -+ 00), and the proofis complete.
o Theorem 3.3 is now applicable, and Theorem 3.1 follows.
Remark 3.6 : It has been conjectured that for the Dirichlet problem (1.1)-(1.2) there will in general exist at most two distinct solutions. The following example which was kindly communicated to me by H. Wente shows that pathologies may occur if the group of symmetries of the data Xo is too large.
=
Example 3.7: Let Xo(u,v) u, 0 < H < 1. By Theorem 1.1 and Remark 1.5.i) the function XH = Xo is the unique solution of (1.1), (1.2) with IIXHllLoo :5 I, which moreover furnishes a relative minimum of DH on Xo + H;·2(BjJ1i3). Theorem 3.1 now implies the existence of an unstable solution X H of ( 1.1), (1.2). The image of XH cannot lie entirely on the Xl_axis: otherwise xi! 1\ X! 0 and l:::..XH = 0 ,i.e. IIXHIILoo:5 1 by the maximum principle, and XH = XH. So XH(w) has a non-vanishing component in direction of the X2_ or X3_axis at some wEB. Rotating X H around the Xl_axis hence generates a continuum of distinct solutions to (1.1), (1.2). It remains an interesting open question whether the "large" solution to (1.1), (1.2) is unique for boundary data which do not admit isometries of J1i3 as symmetries, i.e. which do not degenerate to a line segment.
=
IV. Unstable H-surfaces.
4. Large solutions to the Plateau problem ("Rellich's conjecture"). result analogous to Theorem 3.1 also holds for the Plateau problem:
127 A
Theorem 4.1: Let r be a Jordan curve of class C 2 in IR?, Hi-O, and suppose that DH admits a relative minimum X H on C(r). Then there also exists an unstable solution XH E C(r) of(III.l.l) - (III.1.3).
Remark 4.2. i) Theorem 4.1 for certain "admissible" curves r and sufficiently small H i-0 was established by the author in [3]. Steffen [4] then was able to show that in fact all rectifiable Jordan curves are "admissible" in the sense of Struwe [3]. Independently, and only a few weeks later Brezis and Coron [2] were able to extend their results for the Dirichlet problem and established non-uniqueness in the Plateau problem (III.l.l) - (111.1.3) for r C BR(O) and 0 < IHIR < 1, a result which is optimal when r is a circle. Tlieorem 4.1 was finally established by Struwe [2]. By Theorem 111.3.1, our Theorem 4.1 contains the Brezis - Coron result; moreover, Theorem 4.1 also applies in the case of Theorem 111.3.4 where the method of Brezis and COlOn is not applicable: If we only assume that H2 D(X) < j1r for Borne X E C(r) we cannot guarantee solvability of the Dirichlet problem (1.1), (1.2) for all boundary data Xo E C(r), whereas Brezis and Coron crucially use the existence of H -extensions for all data Xo E C(r). ii) By using results of Wente [2] on the Plateau problem with a volume constraint Steffen [1] in 1972 established the existence of large solutions to (111.1.1) (111.1.3) for a sequence of curvatures Hm -+ O. iii) Theorem 4.1 establishes a conjecture often attributed to Rellich; however, no direct reference is known. We now proceed to set the stage for the - rather tricky - proof of Theorem 4.1. First, however, we state the following a - priori - estimate for H -surfaces which will play a cruical role in our arguments. Theorem 4.2: For any H-surface C IR? there holds the estimate
X
spanning a rectifiable Jordan curve
r
Proof: We present the proof for smooth r, whence X E C 1 (Bj zR3) Theorem 111.5.5. Simply compute, using (111.1.1) - ( 111.1.3):
by
128
B. Surfaces of prescribed constant mean curvature.
0=
J[-~:c+
2HXu /\X.,].X dw =
B
:::;3DH(X) - D(X)
J + JId~ Xl
:::;3DH(X) - D(X)
+ L(r) IIriILoo.
=2D(X)
+ 6HV(X) -
8n X . X do
8B
do ·IIXIIL OO (8B)
8B
Choosing the origin in
1R?
suitably, we can surely estimate
and the theorem follows.
o Now let M =Mt
X
H~,2(Bj IIl?) c
c ({ id} + Tt)
X
H~,2(Bj JR3)
=:T,
and define a map X: M
-+
C(r) by letting
( 4.1)
where
= (:Co, z), :Co E Mt, Z E H~,2(Bj IIl?), Xo(:Co) = h(-y :Co) E Co(r), Z(z) = z E H;,2(B jIIl?). :c
0
Moreover, let EH(:c) := DH(X(:c».
The following lemma is immediate from Lemma 11.2.5. Lemma 4.3:
The map
X extends to a differentiable map of T into
LOO(BjIIl?) + H;,2(BjJR3)j EH extends to a C1-functional on T. As usual we define gH(:C)
=
sup
(dEH(:C),:C - y)
y€M
I.,-ylr
and call a zero of gH a critical point of EH on M. We note
Hl,2
n
129
IV. Unstable H-surfaces.
Lemma 4.4: (III.1.3).
Z
E M
is critical for
EH iff
X
= X(z)
Proof: If X solves (III.1.1) - ( III.1.3) clearly gH(Z) 0 implies that
=
(dD H (X) , tp )
= 0,
gH(Z)
solves (111.1.1) -
o.
Conversely,
u", v.". E Ho1,2(B·,JR3),
and the result follows from Lemma 2.3.
o There is a local compactness condition related to Lemma 3.5:
Lemma 4.5: Zm
= (zmo, zm)
=
Let Xm X(zm) satisfy the conditions
= Xmo + Zm
E Co(r)
+ H;,2(B; JR3),
with
D(Xm)::; C< 00 , DH(Xm ) = EH(Zm) -+ f3 (m -+ 00) , gH(Zm) -+ 0 (m -+ 00) .
Then the sequence {Xmo} is strongly relatively compact in Co(r) and a X(z) E C(r). subsequence {Xm} converges weakly to an H -surface X
=
If
even the sequence {Xm} itself is strongly relatively compact. Proof of Lemma 4.5: Zm
= (zmo,
zm) ~
Z
By Remark 11.2.4 and Lemma 11.2.8 we may assume that
= (zo, z)
weakly, while
Zmo -+ Zo
uniformly on
Similarly, Xm ~ X, Xmo ~ X o, Zm ~ Z weakly, and Xmo
-+
Xo uniformly
in B. Expanding, cpo the proof of Lemma 11.2.11:
Xmo-Xo=d~'Y(zmo).(zmo-zo)-
J J d~2'Y(Z")dz"dZ'
ZmoZmo
~o
where Moreover we note that
z'
2
aBo
130
B. Surfaces of prescribed constant mean curvature.
J
A
A
VZVX dw
= 0,
A
12
A
3
VX E Co(r), Z E Ho' (B;IR ).
B
Hence we obtain (cp. the proof of Lemma 2.2):
2D(Xmo - Xo)
=
J J J
V (Xm - X) V (Xmo - Xo) dw
B
=
VXm V (Xmo - Xo)
+ 2HXmu 1\ X m1l
. (Xmo - Xo) dw + 0(1)
B
=
VXm V (h
(d~-y(zmo). (zmo -
zo»))
B
(h(d~-y(zmo).(zmo-zo»))
+2H X mu I\Xm1l ·
=(dEH(Zm), (zmo - zo») I.e.
Xmo - Xo strongly in
By weak
con~inuity
+ 0(1) :::; gH(Zm)
IZmo -
dw+o(l)
zol + 0(1) - o.
Hl.2(B; IR 3 ), and Zmo - Zo strongly in Mt.
of dDH clearly
Hence
gH(Z) :::;
(dEH(z), Zo - Yo).
sup l'OEMt l~o-eol
But for any Yo E
Mt
Lemma 2.2 and strong convergence Zmo -
(dEH(Z), Zo - Yo)
=
J
on X
(}B
=..E~oo
d~ -y(zo)(zo -
Zo :
Yo) do
J
onXm d~ -y(zmo)(zmo - Yo) do
(}B
m-oo
:::; lim gH(zm)lzmo This shows that
Z
Finally, if DH(X)
X o+H;·2(B;IR3 ).
is critical, or equivalently, that
> {3 -
3~2' we may let Ym
Yol - o. X is an
H -surface.
= Xo + Zm = Xm -
(Xmo - Xo) E Ym satisfies DH(Ym) - {3, dDH(Ym ) - 0 E (H;.2(B;IR3 »)* .
Now we argue as in the proof of Lemma 3.5 to conclude that I.e. Xm - X, and the proof is complete.
Ym -
X strongly.
[]
In order to give the proof of Theorem 4.1. for simplicity we argue indirectly:
131
IV. Unstable H-surfaces.
(4.2)
Auume that there is no unstable H -surface X E c(r).
Let
/30
= inf{DH(X) I X
By Theorem 4.2
is a relative minimum of DH on C(r)}
/30 2: -c(r) > -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)
3}.
Note that (4.2), (4.3) imply the following Lemma 4.6:
/30
X H of DH on /II of JC{3 such that
is achieved at some relative minimum
C(r). Moreover, for any /3 there exists a neighborhood /IInM{3 = 0.
Proof: A minimizing sequence {Xm = X(zm)} of relative minima of DH in C(r) is bounded - by Theorem 4.2 - and hence relatively compact - by (4.3). Any
accumulation point X H by (4.2) must be a relative minimum. By (4.2) again, for any /3 the set JC{3 is both relatively open and trivially closed in M{3. Hence, there exists a neighborhood /II of JC{3 such that /II n (M{3 \JC{3) = 0; a fortiori, /II n M{3 = 0.
o Remark 4.7: In the following we shall apply Lemma 4.6 for /3 < /30+ 3~2 only. Note that by Theorem 4.2 for such /3 we may choose /II c {z E MI IzlT < Ro} where 411" Ro = sup{I:cIT I D(X(:c)) ~ 3/30 + H2 + c(r)}.
An easy modification of the proof of Lemma 11.1.9 also shows the following: Lemma 4.8: Suppose (4.3) is satisfied. Then for any /3 < /30 + 3~2' any E> 0, any neighborhood /II of JC{3, any R > there exists ~ E]O, E[, and a continuous deformation ~: M x [0,1]--+ M such that
°
132
B. Surfaces of prescribed constant mean curvature.
ii)
EH(cJ(:z:, t»
iii)
cJ(Mt'H U {:z: EM 11:z:IT ~ R
t,
is non-increasing in
for any:z: E M.
+ 1}, 1) C Mt'-E UN U {:z: E MII:z:IT
~ R}.
Proof: cJ is obtained by integrating a pseudo-gradient vector field e cut off near the critical set. On a bounded region {:z: E M 11:z:IT ~ R + 1} by (4.3) and bounded ness of X all estimates from the proof of Lemma 11.1.9 convey. Hence (iii)
follows from
lei = IftcJl ~ 1.
(i) and (ii) are standard.
o By Lemma 3.4 given XH C(r) with :Z:l E M and
= X(:Z:H)
with
:Z:H
E Jet'o we can find
Xl
= X(:z:I) E
Moreover,
p = {p E Co ([0, 1); M) I p(O) =
(4.4)
:Z:H,
:Z:l}
#0
E C(r)
,
P(1) =
and
/3H
= pEP inf sup EH(:Z:) < /30 + 4H1I'2 "'Ep 3
More generally, for any relative minimum convexity of M
p=
X = X(:il)
{p E Co ([0, 1); M) Ip(O) =
and we may let
fj
:il,
•
P(1) =
:Z:l}
:il EM, by
#0
= in( sup EH(:Z:) . pEP"'Ep
For such an
X
and
R > 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:») <
-min{~ gk(:Z:),
I},
which we may construct according to Lemma 11.1.8. Let
~: V(~) C M6 0
X
[0,1]-+ Mlo
be a maximal flow of integral curves of e, cpo Lemma 11.1.9. Note that for any Po} the set M6 is forwardly invariant under ~, i.e.
C < min{co,,B -
( 4.17)
~
(MI,t) c M"
"It
~
O.
139
IV. Unstable H.surfaces.
Hence for such fJ clearly 'D(~) J But by (4.15), (4.16) for any z E
M6
X
[0,1].
M6
I (dEH(~(Z,t», 1
EH
(~(z, 1»)
=EH(Z) +
e(.») dt
o
30 + fJ 62
-
min{~ fJ~, I} < /30
if fJ < =to In view of (4.17) this contradicts the definition of follows.
/30' and the theorem
o Remark 4.14: Inspection of the above proof shows that (4.14) and Claim 2 remain true if H2 D(X) ~ ~1r. By uniform boundedness of the volume V(X(z», z E M, cpo Claim 1, DH(X(Z» is uniformly continuous as a function of H on M. Hence the estimate /30 < P of Claim 2, and thus also Theorem 4.13 will stay true for a sligthly larger range of curvatures H.
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