On the size of a stable minimal surface in R3

Author(s): J. L. Barbosa and M. Do Carmo Reviewed work(s):

Source: American Journal of Mathematics, Vol. 98, No. 2 (Summer, 1976), pp. 515-528 Published by: The Johns Hopkins University Press

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By J. L. BARBOSA* and M. DO CARMO.*

1. Notations and results.

1.1. Let M be a two-dimensional, orientable C '-manifold. A domain D C M is an open, connected subset with compact closure D and such that the boundary aD is a finite union of piece-wise smooth curves. Let x: M->R3 be a minimal immersion into the Euclidean space R 3. It is well known that D is a critical point of the area of the induced metric, for all variations of D which keep aD fixed. When this critical point is a minimum for all such variations, we say that D is stable. The goal of this paper is to estimate the "size" of a stable minimal immersion and the main theorem is as follows. Set S =

{

e

R 3; X2 + y2 + z2 = 1} and denote by g: M-> S'2, the Gauss map of the immersion x.

THEOREM 1.2. Let the area of the spherical image g(D) c S' of a domain

D c M be smaller than 27. Then D is stable.

This estimate is sharp, as can be shown, for instance, by considering pieces of the catenoid bounded by circles C1 and C2 parallel to and in opposite sides of the waist circle CO. By choosing C1 close to CO and C2 far from CO, we may obtain examples of unstable domains whose spherical image has area larger than 2v and as close to 27 as we wish. Further details will be given in Section 2.

Since g may cover g(D) more than once, Theorem 1.2 implies (but it is stronger than) that if the total curvature is smaller than 27, then D is stable. Let N be a unit normal field along x(M). Let A = _{div grad and K denote}

the Laplacian and the Gaussian curvature of M, respectively, in the induced metric. Given a piece-wise smooth function u:D--R, with u_0 on aD, the second derivative of the area function for a variation whose deformation vector field is given by V= uN is (Cf. 3.2.3 of [9]).

I(V,V)= U(-/Au+2uK)dM. (1.3)

Copyright ? 1976 by Johns Hopkins University Press.

515

Manuscript received October 17, 1973; revised May 28, 1974. *Partially supported by C.N.Pq and N.S.F.

where dM is the element of area of M in the induced metric. If I (V, V) > 0, for all such V, then D is stable. We say that D is unstable if for some V, I (V, V) < 0. A Jacobi field in D is a normal field uN, where u: D->R is a smooth function which satisfies

-Au +2uK=0. (1.4)

A boundary aD of a domain D c M is a conjugate boundary if there exists a non-zero Jacobi field on D vanishing on aD; if, in addition, there exists no domain D' c D, D' # D, such that MD' is a conjugate boundary, aD is called a first conjugate boundary. The multiplicity of a conjugate boundary aD is the

number of linearly independent Jacobi fields on D vanishing on MD.

Theorem 1.2 is related to some results of A. H. Schwarz (see [8]). In Section 2 we prove these results in our context. We also give a simple proof of the fact that the multiplicity of a first conjugate boundary is one.

In Section 3 we prove Theorem 1.2 and indicate another application of the ideas of the proof. The section closes with a few open questions.

We want to thank S. S. Chern for having suggested this question to us. Thanks also due to R. Osserman, who read critically a preliminary version of this work, and to R. Gulliver, who pointed out some gaps in our first proof. Conversations with J. Cheeger, S. Y. Cheng, J. Kazdan, B. Lawson, J. Simons, N. Wallach and F. Warner were helpful during the preparation of this paper.*

2. A result of A. H. Schwarz

2.1. Let W be a two-dimensional real analytic Riemannian manifold and let D c W be a domain in W. We will denote by A w the Laplacian of W and by H (D) the space of C ? functions on D which are not identically zero and vanish on MD. A real number X >0 such that there exists a solution of AWiu + Au

= 0, u E H (D), is called an eigenvalue in D for A w (this is actually the negative of the usual eigenvalue). The space

PA (D) = (uEEH(D);Awu+Au=0}

of such solutions is the eigenspace corresponding to A. It is known that if u E PA (D), u is analytic in D. It is also known that the eigenvalues in D form a discrete set of positive numbers, and, as usual, we order then so that

n,1<\2< ''' -</n' -

We also denote by _{fI lI} the norm associated to the Riemannian metric <, > of W, by dW the element of area of W, and by H'(D) the set of functions u in D which are not identically zero, vanish on aD and are such that u2 _{and IIgrad} U 12 are integrable on D.

LEMMA 2.2. For all u EH1(D),

flgradu l12dW

fudW

and equality holds if and only if u _{E PA1(D).}

For a proof see [1] p. 186.

The following lemmas are known. Since we have been unable to find explicit references applicable to the present case, we provided short proofs.

LEMMA 2.3. If D c D then X1 > X1, and equality holds if and only if D = D (here X1 is the first eigenvalue in D).

Proof. Let u E H '(D) and extend it to an element of H '(D) by setting u -=0 outside D. This shows that H'(D) c _{Hl(D) and, by Lemma 2.2, j > Xl.} Assume now that X1 = _{X1 and let ii (EPX,(D).} Define u E H'(D) by setting u = ii in D and u =0 in D _{- D. Then we have fJ-IJgrad U112dW=Xf-Ju2dW,} and, by Lemma 2.2, uEP/AJD). Therefore u is analytic in D, and D- D can have no

interior points. Q.E.D.

LEMMA 2.4. Let LAw u + Xu = O, u E H (D). Then X = X1 if and only if u 0 in D.

Proof. Assume that X=Xj. By a known property of solutions of elliptic equations ([4], p. 210) it follows that if u( p) =O,p ( D, then u changes sign in D. This implies that there exists a domain D' c D, D' # D, such that u restricted to D' satisfies A?4Wu +Xu = 0 and u vanishes on WD'. Since ([1], p. 121; there is a difference in sign from our Laplacian to the Laplacian in [1])

Algrad

U112

dW=

uAwudW=X1fu2dW,

D' I2 D

we have, by Lemma 2.2, that the first eigenvalue X' in D' satisfies X' < _Xl. On the other hand, by Lemma 2.3, XA _{> Xl, and by the second part of Lemma 2.3,} D = D'. This contradiction proves that u: 0 in D.

([1], p. 121)

fuAwuidW= - <gradu,gradu1>dW= fu,AwudW, _(2.5)

we obtain

(A 1) u1 dW= 0.

Since A 7AX and, by the first part of the lemma, uj1 0 in D, we conclude that u changes sign in D. This contradiction proves that A = _A1. Q.E.D.

2.6. We now return to the situation of Section 1 and assume that M is not a plane. If we look upon M and S2 as Riemann surfaces, the Gauss map g: M_> S2 is known to be antiholomorphic. It follows that K can only be zero at a countable number of isolated points Pl, *.. * Pk, .... Set S = _{M-{ P1} ... Pk ... }

Restricted to S the Gauss map is an immersion; hence it can be used to define a new Riemannian metric <, >s on S by setting

<UV>s =<dg(U),dg(V)>5 U,VET(S),

where <K> denotes the Riemannian metric induced in M by R 3, dg is the differential of g, and T(S) is the tangent bundle of S. We will denote by As and dS the Laplacian and the element of area, respectively, in the metric <K >s.

LEMMA 2.7. Let D be a domain in M. Let u EH(D) and let ii be its restriction to D n S. Let V= uN. Then V is a Jacobi field in D if and only if

As5i + 2ii = O. (2.8)

Furthermore

I (V, V) f _(iAsii + 2i2 ) dS. (2.9)

Proof. By using the fact that x(M) is minimal, we can easily check that

<dg(U),dg(V)>=K<VuN,VvN>= -K<U,V>,

We can now describe, in our context, a result of Schwarz. For the notion of (global) branched covering, we refer to [2] pp. 220-221.

THEOREM 2.7. (Schwarz [8]). Let D c M be a domain and assume at the Gauss map g: M_S 2, restricted to D, is a branched covering onto g(D). Assume further that the first eigenvalue X1 for the Laplacian As in g(D) c S 2 is smaller than two. Then D is unstable.

Proof. Let ucP1( g(D)), i.e., A8 +Xu =O and u-0 on a(g(D)). Since = g restricted to D is a branched covering, g(aD) = a (g(D)). The function f= wg: D-1R satisfies _Asf+ X1 f=0 in Dn S, is positive in D, and vanishes on

MD. Set V=fN and let f be the restriction of f to D n S. Then

I(V,V)=- f(fsf+2f2)dS=X ff2dSN2ff2dS. Dnr S(si Df)dS=X n s

d-

Dns f

Since X1 <2, we obtain for such a V that I (V, V) <0. Thus D is unstable. Q.E.D.

Schwarz's theorem is more widely known through one of its consequences, which can be described as follows. Let U: Sj2->R be the restriction of the coordinate function u (x, y, z) = _{z to S2. It is easily checked that A u + 2u} = 0 (u is a spherical harmonic of first order). Since u > 0 on the hemisphere H ={(x,y,z)ES2;z> 01 and vanishes on M, we conclude by Lemma 3 that the first eigenvalue in H is two. From Schwarz's theorem and Lemma 2.3 it follows that (cf. [7]): if the Gauss map is 1-1 and D c M is a domain such that g(D) covers a hemisphere, then D is unstable.

There are other domains of the sphere which are known to have two as first eigenvalue for the Laplacian. For instance, it is easily checked that the restriction to S2 of

(l+z)

uC,Z(xYz)=2-zlog (1_Z) +cz,z&1,0S c <oo

is a solution of Asu + 2u = 0; u is positive in a ring-shaped domain C bounded by two parallels, and vanishes on aC. Thus the first eigenvalue of C is two and Schwarz's theorem can be applied, using C instead of H.

one of the boundary parallels of C approaches the equator while the other approaches a pole. The area of the spherical image is then 2X. For details on the catenoid see [5].

Linear combinations of uC u, UC uC x yield further domains which have two as the first eigenvalue for the Laplacian.

We finish this section by giving a simple proof of the following fact:

PROPOSITION 2.11. The multiplicity of a first conjugate boundary of a minimal immersion x: M->R 3 is one.

Proof. Assume that D c M and that aD is a first conjugate boundary. Let

yt be the vector space of solutions of - Au + 2uK = 0 vanishing in aD. Fix p E D and define a linear map q :pl-1R by p(u) = u(p). We claim that u(p) = 0 if and only if u is identically zero; because if u

(

([4], p. 210) and aD is not a first conjugate boundary. Thus, 9p is injective. Since u (p) #0 for some u, we have that qp is an isomorphism. Therefore dim tt = 1. Q.E.D.

3. Proof of Theorem 1.2 and some comments.

3.1 We will assume that D is not stable and will arrive at a contradiction. If D is not stable, I (V, V) < 0, for some V= aN, where a: D-I>R is a piece-wise smooth function vanishing an aD. By the Morse Index Theorem [10], this means that D contains a domain D' (which may be D itself) such that aD' is a first conjugate boundary. Thus, there exists a Jacobi field J = uN vanishing on aD', that is, u is C?? on D', satisfies -Au+2uK=0,u>0 on D' and u_O on dD'. Under these conditions, we will construct a function f:g(D')-*R such that:

fEH'(g(D')) and f _gradf1l2dS<2f f2dS. (3.2)

g(D') g(DI)

The existence of such a function implies, by Lemma 2.2, that A1( g(D')) < 2. On the other hand, it is a known fact (see [6] p. 19, or [3]) that among all spherical domains with the same area, the spherical cap minimizes the first eigenvalue for the Laplacian. Let C be a spherical cap with

area C= area g(D') < 2,g.

conclude that

X1(g(D')) >A1(C)>2.

This contradicts our previous inequality and proves Theorem 1.2 modulo the construction of f.

3.3. To construct f, we start with the following observations. Since the Gauss map g is anti-holomorphic, the curvature K can only be zero at isolated points in M. In a neighborhood of each such point (to be called a branch point), g is a branched covering with finitely many sheets; the number of such sheets is called the multiplicity of the branch point. By compactness, there is only a finite number of branch points in D'.

Of course, if p E M is not a branch point, g is a local diffeomorphism in a neighborhood of p and we may say that its multiplicity is one. It will be convenient in what follows to associate to each p E M a multiplicity a > 1.

Now let qEg(D'), let g'-(q)rnD'={p,...,pp}, and let a,>I be the multiplicity of pi, i = 1,..., nq. Set

f ( q) Eai u ( Pi). (3.4 )

Since u>O on D' and u_O on AD', the function f:g(D')-*R defined by (3.3) is positive on g(D') and vanishes on a (g(D')). The whole point of the proof is to show that f satisfies (3.2).

3.5. To prove (3.2), we will need a series of lemmas. It will be convenient to extend the function u to M by setting u_ 0 outside D'; this extension is still denoted by u.

LEMMA 3.6. f is continuous in g(D').

Proof. Let q E g(D') and use the notation of (3.4). There exists a neigh- borhood U c S' of q such that the set formed by the connected components of g '( U) which meet D' is the disjoint union of neighborhoods U1,..., Un C M of the points pl...., Pnfq respectively, and the restriction of g to Ui -{ pi} is an ai-sheeted covering of U -

{

fi (4) U

a( pji) j

ai,

We may assume U to be a small geodesic disk of S' around q. Let L be a geodesic segment joining q to au. Then g has inverses:

g1: :U-L-* Ui

and

fI U-L =u?g7'.

Since u _ 0 outside D', each fi is continuous on U - L. Since L is arbitrary, fi is continuous in U - _{{ q}. But, for any choice of L, we have that}

lim Eu?gi1 i iu( Pi )

and this shows that fi is continuous in U. Q.E.D.

3.7. Let {ql,.. q be the set of points in g(D') that are images of branch points in D'. Let Be', .. ., Btm be pairwise disjoint closed geodesic disks in S' of radius e about ql,... q, respectively. Set

TI=g(D')-g(aD')-

U

t

LEMMA 3.8. It is possible to find simply-connected disjoint open sets

Rik C T, such that:

(i) if q E _Rik then g - _{l(q) n D' has exactly j elements.}

(ii) TE= U ,kRik

(iii) g -(R,k)= U Ii= R,,l, where R,kl are disjoint open sets and g restricted to

R,kl is a diffeomorphism onto R,k.

Proof. Set

T=

{

Since g is a local diffeomorphism on g '(TE)n qD', T is open. Let Tk5 k=

1,... _{,n, be the connected components of li. If some T is not simply-con-} nected, it is possible to cut it along curves to obtain a simply-connected set R,k.

It is easily seen that T? = U _Rik, and this proves (i) and (ii).

restricted to R,kl is a covering map. Since

Rik is simply-connected, so is R,kl, and

glRikI is a diffeomorphism onto Rik. This also shows that there are j connected

components in g (Rik)nD'. Q.E.D.

Remark 3.9. The boundary adRk is a union of arcs ny1, where y nis an arc of aBet or an arc contained in g(aD') or an arc of one of the curves used in the proof of Lemma 3.8 to obtain simple connectivity. This remark will be useful later.

LEMMA 3.10. Let U be an open set of M where g is a diffeomorphism. Then

11 d (u og-1)112=- KIddui2.

Proof. Let el, e2 be an orthonormal frame around a point of U and wl, w2 be its dual coframe. The second fundamental form is 11= E hi.wiw. ij=1,2, where h12 = _{h2l and, by minimality, h1l + h22} = _{0. Furhtermore, the matrix of dg}

in the basis {el,e2} is _{dg=(hij), and K=det(h21).}

Now set du = a1w1 + a2w2. Then

K 2j1 d (u o g - 1) -12= K2lIdu o dg 'l2s

= (alh22- a2h12)2+ (- _alh2l - 2h22)2

=(a + a2)(h21 + h2)= -Kjddujj2.

Q.E.D.

3.11 We now start proving (3.2). It will be convenient to set g,kl = gIR1kIl By (iii) of Lemma 3.8, gikl has an inverse

gl.kll : k , jk15

which can be extended, by continuity, to R,k. Observe also that

fi Fk EU ? gikl5) I=1 ,j

The idea of the proof is to show that

lim f jjdfII2dS< limr2ff dS.

Since the right hand side of this inequality is bounded, because f is continuous, so is the left hand side. Thus f E H '(g(D')) and, in the limit,

/(') j1 dfjj2dS < 2 f2dS,

g(Dt) ~~~g(fl')

as required.

We first compute fT 1dfjj2dS.

f _Tdf2ds1 Jdfjj2dS= ₁₁ E f

11

= I

where

I

E

J 2

E<d(u og-kr),d(u og-'5)>dS, r,sl S. ,j.5

jk Rk r<s

By using Lemma 3. 10 and the fact that dS =-KdM, we obtain

E

_ldud2dM.

By setting _{D, = g'- 1(u,B,),} we have D'- D= U R,,kl. Thus

f1 df 2ds=

d

I

f D I _IdUII2dM+ I (3.12)

On the other hand,

ff2 dS = f( U?gjI1)2 dS

-

E

?g,1')2 dS+ II,

where

11=

,

that is,

2ff2 dS= 2E2 f Ku2 dM+ 211

jkl Rjkl

=-J_D 2Ku2dM+ 211. (3.13)

U2=0

Now we notice that uAuu-2Ku2=0 on D'. Thus, by using (3.12), (3.13) and Stokkes theorem, we obtain

fIdf 12dS

-

I - 2II

= J <ugradu,N>ds+I-211

aDf

where N is a unit normal vector to aDf and ds is its element of arc. Since u is differentiable in D' and vanishes outside D',

lim f _{<ugradu,N>ds=O,}

E - _DIE

and we are reduced to prove that limE yo(I - 211) < 0.

3.14. It will be convenient to set

v 8gjkl :jR =,s

Observe that fl Rk= v . _{Furthermore, from the fact that} Au -2Ku 0, it follows that Asv' + 2vL= O, hence

AS(Vr.Vs) = V rAVs, + _VsASV r+ 2<dVrdvs> =-4vrvs+2<dv rdvs>.

Therefore,

I-211= t _(2<dv rdvs>-4vrvs)dS

jk r<s Rik

where aR,k is positively oriented, N is a unit normal vector to aR,k pointing outwards, and da is the element of arc of aR,k.

Now we observe that if v vS=0 along some arc a of adRk then, since v vS > 0 on Rjk,grad(vvrS) points inwards (or is zero) along a. Thus <grad(vv S), N> < 0, and the contribution of a to the above boundary integral is

non-positive. Furthermore, by taking into account Remark (3.9), if v v'S 7 0 on some arc y of aR,k, then either y is a common boundary of two such regions, in which case it appears twice with opposite signs in the above integral, or y is an arc of aBMt, for some t. Thus we are reduced to prove that

lim

_E

where -yink is an arc of some aBt.

By using the fact that u < M on D', we obtain

f<grad(vrvs),N>da = <vrgradvs+vsgradvrN>dca

I

fvrl I<gradvs,N> da+

+

Yjk Y,k

SMf (<gradvs, N>I + I<gradv r5 _N>I) da.

Yjk

Now, let -jn be an arc of aBM (we will drop the superscript t), let q be the center of

_B,

f<grad v r,N>da= dv r(N) da= fd(u og,rl ) (N) da

=ftn(duo? _dg~r1k)(N) da =du( K

)

=

The same proof yields the following companion of Schwarz's theorem:

PROPOSITION 3.17. Let D cM be a domain and assume that the first eigenvalue A1 of g(D) c S2 is greater than two. Then D is stable.

In particular, if g(D) is contained in the closure of one of the domains referred to at the end of Section 2, D is stable.

We conclude with a few remarks and questions.

3.18. It is possible to find on the catenoid domains which have spherical images with area larger than 27 and yet are stable. This raises the question of improving the estimate of Theorem 1.2 by introducing further geometrical quantities associated to D.

3.19. For jninimal hypersurfaces x: M'-*R ln+l of Euclidean space R n+1 (superscripts denote dimension) the Jacobii equation reads Au+ _IIA112u = 0, where A is the Laplacian of M in the induced metric and IIA I is the norm of the second fundamental form A (see [9]). Theorem 1.2 implies that (n = 2) if a domain D c M satisfies f-I I A112dM <47 then there exists no solution of the above equation which vanishes on aD. Can a similar estimate be obtained for n > 3? The case n = 2 was made possible by the fact that the Gauss map is anti-holomorphic.

3.20 For a minimal immersion x: M2-R Rn+2 the generalized Gauss map is a holomorphic curve on a complex hyperquadric Qn of the complex projec- tive space pn+ '(C). Can the stability of x be described in terms of this holomorphic curve? When n=2,Qn is the product S2XS2 of two spheres, looked upon as P l(C) x P l(C); this circumstance might make this case more manageable.

INSTITUTO DE MATEMATICA PURA E APLICADA.

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