Plateau's Problem and the Calculus of Variations

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
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