Stability of minimal surfaces and eigenvalues of the Laplacian

Math. Z. 173, 13-28 (1980)

Mathematische

Zeitschrift

9 by Springer-Verlag 1980

Stability of Minimal Surfaces and Eigenvalues

of the Laplacian

J05.0 Lucas Barbosal and Manfredo do Carmo 2

Departarnento de Matematica, Universidade Federal do Cearfi, Fortaleza Cearfi, Brasil 2 Instituto de Matematica Pura e Aplicada, Rua Luiz de Cam6es 68,

20060 Rio de Janeiro R.J., Brasil

1. Introduction

(1.1) Let x: M ~ / " be a minimal immersion of a two-dimensional orientable manifold M into an n-dimensional Riemannian manifold ~/". Let D c M be a domain in M with compact c l o s u r e / ) and piecewise smooth boundary ~D. D is then a critical point for the area function of the induced metric, for all variations of 15 keeping 0D fixed. We say that D is stable when such a critical point is a relative minimum.

In a previous paper [1] we proved that if ~ / " = R 3 and the total absolute curvature of D is smaller than 2~r then D is stable (actually we proved a stronger result). In the present paper we extend this result in various directions. Ex- plicitly, we prove the following results. Let K denote the Gaussian curvature of M in the induced metric.

(1.2) Theorem. Let x: M--+S3(a) be a minimal immersion of M into the three- dimensional sphere S3(a) with constant curvature a > 0 . Assume that D ~ M is simply-connected and that

~ ( 2 a - K ) d M <2rc. D

Then D is stable. Furthermore the result is sharp in the following sense: given 6 > O, there exists a minimal immersion x: M ~ s a ( a ) and an unstable domain D 6 c M

such that

( 2 a - K) d M = 2rc + 6. D6

(1.3) Theorem. Let x: M--+ H3(a) be a minimal immersion of M into the hyper- bolic space Ha(a) with constant curvature a < 0 . Assume that D c M is simply- connected and that

~ [ K l d M < 2 m Then D is stable.

14 J.L. Barbosa and M. do Carmo

(1.4) Theorem. Let x: M--*R n be a minimal immersion of M into the n-dimen- sional euclidean space R". Assume that D ~ M is simply-connected and that

S IKf

D

Then D is stable.

(1.5) Remark. It has been proved by Spruck 1-20] that there exists a n u m b e r ~(n), depending on n, such that if the total absolute curvature of x : D ~ M - - , R ~ is smaller than ~(n), D is stable. Our explicit bound in Theorem (1.4) does not depend on n. However, in constrast to the case n = 3 of [13, we do not know whether the bound (4/3)n is sharp. It is an interesting question to decide whether a better bound such as 2re can be achieved (2n would then be a sharp bound; see 5.1).

(1.6) Remark. The stability of domains D ~ M with ~ I K l d M < 2 n in minimal

D

immersions x: M---~R 3 was used by Nitsche [12] to prove that if F ~ R 3 is an analytic curve with total curvature smaller than 4n, then F bounds a unique solution to the Plateau problem. Theorem (1.3) raises a similar question for curves F c H S ( a ) satisfying an analogous condition. By a result of Kaul ([9], p. 205) such a condition o n

F~H3(a)

implies, as in the case of R 3, that a solution x(D) to the Plateau problem with x ( ? D ) = F has no branch points and ~ [ K l < 2 n . Thus such a solution is stable by Theorem 1.3. If one can prove a D

version of Schiffman's theorem [18] in H3(a) for variational minima, then Nitsche's theorem will extend to this case.

(1.7) Actually what we present in this paper are some examples of a method to obtain sufficient conditions for stability of minimal surfaces of Riemannian manifolds with bounded curvature. The method can be applied successfully when the ambient Riemannian manifold is either R", SS(a) or HS(a). It can also be applied when Mn is a space of constant curvature, but the results are not so interesting and will be presented elsewhere.

Roughly speaking, the method can be described as follows. The assumption that D is not stable leads, through the second variation formula, to an upper bound c~ of the first eigenvalue 21 of a problem of the form

(1.8) A f + 2 F f = O , in D ' ~ D , f = 0 in 0D'.

where F is a nonnegative function on M which only vanishes at isolated points, and depends on the curvature K of M and on the curvature bounds of the ambient space. We change the induced metric ds 2 of M to a (possibly de- generate, see 3.12) metric d~r 2 = F d s 2, and show that the Gaussian c u r v a t u r e / ( of d a 2 is bounded above by a n u m b e r K o > 0 which can in m a n y cases be estimated in terms of the dimension and the curvature bounds of the ambient space. We then show, by a standard symmetrization procedure, that 21 > 2 l (D*), where ,~I(D*) is the first eigenvalue of the Laplacian of a geodesic disk D* in a sphere with constant curvature K o so chosen that

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 15

Since we can estimate LI(D* ) as a decreasing function of the area of D* (see Sect. 3), we can choose a constant A so that the condition ~ F d M < A implies

that D'

C t ~ 2 1 > ) ~ I ( D * ) > ~.

This is a contradiction, and shows that a condition of the above type implies stability for D.

This method may yield a general theorem for stability of minimal surfaces in Riemannian manifolds. The function F is easy to be determined; however the matter of estimating /s is far from being completely settled and we will not pursue the subject further in this paper. We rather concentrate on the above mentioned special cases where explicit and interesting bounds can be obtained.

In Sect. 2 we present the estimates o f / ~ needed for the proof of Theorems (1.2), (1.3) and (1.4). In Sect. 3 we describe how to estimate the first eigenvalue of the problem (1.8) by the first eigenvalue of the Laplacian of geodesic disks in spheres; this Section can be read independently of the rest of the paper. In Sect. 4, we prove Theorems (1.2), (1.3) and (1.4). Finally in Sect. 5, we present some final remarks.

We want to thank R. Gulliver for pointing us out a mistake in a first version of Theorem (1.4).

2. Curvature Estimates

(2.1) Let x : M ~ A 3 ( a ) be a minimal immersion, where A3(a) denotes the simply-connected three-dimensional space of constant curvature a, a a real number. Denote by ds 2 the metric on M induced by x and by K its Gaussian curvature. It is known that if a = 0 then the Gaussian c u r v a t u r e / s of the metric

da2= - K d s 2 satisfies/~-= 1. F o r an arbitrary a, we extend this result as follows.

(2.2) Proposition. The Gaussian curvature Is of drr2= uds 2, where

u = 2 a - K , /f a > 0 , (2.3)

u = - K , /f a < 0 ,

satisfies Is < 1.

Proof. Choose an adapted frame et, e 2, e a to the immersion x: M~A3(a) and let c01, e) 2 be the dual coframe to el, e 2. Let hu, i , j = l , 2, be the coefficients of the second fundamental form of x in the frame el, e2, and set ] [ B l l 2 = ~ h 2. The

ij Gauss equation implies that 2 a - 2 K = [/B]I 2. Thus (2.3) can be written as

if a > 0 , u=a+ 89

(2.3)'

if a < 0 , u = - a + 8 9

It will be convenient to set a § = m a x ( a , 0) and write the above as

16 J.L. Barbosa and M. do Carmo

Now it is easily computed that (see e.g. [-3], Eq. (2.8)) the Gaussian curvature

Is of da2=uds 2

is

where K is the Gaussian curvature of

ds 2, A u

is the Laplacian of u in the metric

ds 2

and

ul, u z

are given by

du=u lco~+u

2o) 2 .

Since u = 8 9 hZ+lal, we obtain (see [7])

U k = 2 h i j h i j k , A u = Z h 2 k + ~ h q A h l j ,

i,j,k=l,2.

ij ijk ij

It is known that ([7], p. 38)

A hij = - hlj(~ h2l) + 2ahij,

kl hence

A u = Z h~k-- (~ (h,j)2) z + 2a Z (h,j) 2

ijk ij ij

= 2 h/~k-- (2u-- 2 [al) 2 + 2a(2u-- 2 [al). ijk

It follows that (2.5) can be written as

(2.6) /~ 2a+u-U

+_2@{(2u_21aj)2_2a(2u_2[ar)}_~ula

1

+ ~u3 {- 89

~rs h2 ~ h~'k + ~k

(2ij

hijhijk)2}"

h~k

ijk

We first observe that the last summand of (2.6) is zero. To see that, we set

H= 1 Z h~Z~ Z h~k + Z (Z hijhijk) 2,

rs ijk k ij

and choose the frame el, e 2 so that

hi2-~h21 =0;

this choice can be made except at a finite number of points, and the final result is obtained by continuity. With such a choice,

I 2 2 1 2

H=--ghll(~,hqkl--~h22(Z hek)+(hllh111

+h22h221) 2

ijk ijk

+ (hi 1 h122 + h22

h222) 2.

Since, by minimality, h l ~ + h22 = 0 and hlj k is symmetric in all indices ([7], p. 37), we obtain

H = _ h 2 (4h~1z +4h2 2)+ah21h211

_{+ 4 h l l h 1 1 2 = 0 ,} 2 2

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 17

Thus from (2.6) we obtain

(2.7) /( < (2a+ - u) u + 2(u - l a l) z - 2a(u -lal)

z /./2

Now let us assume that a > 0. Then

u 2 - 4 a u + 4 a 2

/ ( < u2 =f(u), u>a.

Notice that f ( a ) = l , f ( 2 a ) = 0 , l i m f ( u ) = l . Furhtermore, by looking at the derivative f'(u) we see that ,4oo

f ' ( a ) < 0 , f ' ( u ) = 0 if and only if u=2a,

and f ' ( u ) > 0 if u>2a. It follows that / ( < f ( u ) = < l and this proves the pro- position for a > 0.

If a < 0 , we obtain from (2.7),

/( < - 1 +u2~ (u 2 - 2 u l a l + lal 2 --au + alal)

= --1 -t-~ff(u2q-au)=l + 2 a < l u

and this concludes the proof of Proposition 2.2.

(2.8) We now consider the case of minimal immersion x: M ~ R " . Let ds 2 be the induced metric and set do -2= - K d s 2. In principle, we could find an upper bound for the curvature /( of &r 2 by using the method of Proposition 2.2. A better bound however can be obtained by the following observations (see [8] for details).

The Gauss map ~b: M-~II;P "-1, of M locked upon as a Riemann surface, into the complex projective space C P " - 1 defines M as a holomorphic curve in the hyperquadratic Q , - z ~ CP"-1. By normalizing the Fubini-Study metric in C P " - 1 so that the maximum seccional curvature of C P " - x is 2, it can be shown [8] that the metric induced by q~ on M is precisely d o - 2 = - K d s 2. Since a holomorphic curve is a minimal surface, we obtain:

(2.9) The curvature I( of da 2 satisfies I( < 2.

(2.10) Remark. The b o u n d / ( < 2 is sharp for n > 3 . For instance, one can easily c o m p u t e / ( for the minimal immersion in R4:

X(U, V)=(U 2 --V 2, - 2 u v , - 2 v , 2u) and o b t a i n / ( - 2.

18 J.L. Barbosa and M. do Carmo

3. Eigenvalues of the Laplacian

(3.1) This section can be read independently of the rest of the paper. Let M be a two-dimensional C2-manifold endowed with a C2-Riemannian metric ds 2. Let D c M be a domain in D with compact closure /5 whose boundary ~D is a piecewise Cl-curve. We will denote by A the Laplacian on M and by H(D) the space of C2-functions o n / 5 which are not identically zero and vanish on 0D. We will denote by 21(D ) the first eigenvalue of A in D, that is, 21 is the smallest real number such that there exists a solution of

(3.2) Au+21u=O, u~H(D).

We will also denote by dM the element of area of M, by Vu the gradient of the function u, and by HZ(D) the set of functions u in/5 that are not identically zero, vanish on 0D and are such that u z and II gull 2 are integrable on/5. It is known that

II

Vul[ 2 dM

21(D) <D , u~H~(P), u2 dM

D

and equality holds if and only if u is a solution of (3.2).

Finally, we will denote by K the Gaussian curvature of M, and will write

M(Ko) to indicate that M is a surface with constant Gaussian curvature K o, where K o is an arbitrary real number.

The argument of the following Proposition is essentially due to Peetre [15] who considered the case K o - - 0 (actually, it has been pointed out to us by Osserman that the Proposition itself is contained in Bandle [4]). Since we need parts of the argument for the next Propositions, we present the proof here for completeness.

(3.3)

Proposition.

Let D ~ M be simply-connected and let the Gaussian curvature K of M satisfy K<=K o. Let D * ~ M ( K o ) be a geodesic disk in M(Ko) such that

area D* = area D. Then

(3.4) 21 (D) >= 21 (D*)

and equality holds if and only if D=D*.

Proof. Let ueH(D) be a solution of (3.2). It is known that u > 0 in D. By a standard symmetrization procedure ([16], pp. 100-104) set

D(r)={pED;u(p)>r}, 0 _ < r < m a x u ,

(notice that D(0)=D) and consider concentric geodesic disks D*(r) in M(Ko)

such that

A* (r) = area D* (r) = area D (r) = A (r).

Define a function u* on D*(O)cM(Ko) by requiring that u*=r on OD*(r) (thus the geodesic circles ~D*(r) are the level lines of u*). Now set

G(r)= ~ I[Vull2dM, L(r)= ~ ds, H ( r ) = ~ u2dM,

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 19

where

ds

is the element of arc of ~D(r); denote the corresponding quantities in D*(r) by a star. It is known that ([16], p. 100)

G ' ( r ) = - ~

IIVullds,

A ' ( r ) = - j"

IIVull-lds,

OD(r) OD(r)

where prime denotes the derivative with respect to r. Thus, by Schwarz's inequality,

(3.5) (L(r)) 2 =<

G'(r) A'(r).

We now use the isoperimetric inequality for surfaces with curvature bounded by K o (see [5] or [2]):

(3.6) (L(r)) 2

>=4hA(r)

(1

A(r)4rc

K~

equality holds in (3.6) if and only if

D(r)

is a geodesic disk in

M(Ko).

Since u* is a function of r alone and

A*(r)=A(r),

G*'(r) = - II Vu*

II L* (r), A'(r) = - II

Vu*

II -1 L* (r), hence

(3.7) (L* (r)) 2 = G* '(r) A' (r).

F r o m (3.5), (3.6) and (3.7), we obtain, for each r,

(3.8)

G*' A'=(L*)2=4n A (1-A4Kn~ ) <_ L2 <_ G' A ',

hence, by integration on r, 0 _< r _< max u,

G*<=G.

Since,

H*(r)=H(r),

we obtain, at r = 0 ,

(3.9) 21(D) = G(0) > G*(0) > 21(D, ) H (0) = H* (0) - "

Equality holds in (3.9) if and only if it holds in (3.5), (3.6), and u =u*. Thus D is a geodesic disk in

M(Ko).

q.e.d.

The method of proof of Proposition (3.3) can also be used to obtain lower bounds for the first eigenvalue of spherical domains. Let us denote by

S2(R) a 2-

sphere with radius R. Then we can obtain the following estimates.

(3.10) Proposition.

Let

D~S2(1)

be a simply-connected domain with area A.

Then

i)

If A<2n, 21(D)>=4n/A.

ii)

If 2n<_A <4n, 21(D)> 2(4n- A)/A.

Proof

i) Let R be such that the total area of S2(R) is 2A. By Proposition (3.3), 21(D)>21(D*), where

D* cS2(R)

is a geodesic disk with area A. But then D* is a hemisphere H of S2(R), the first eigenvalue of which is easily seen to be

2 4re 2 1 ( H ) = R 2 - A

20 J.L. Barbosa and M. do Carmo

ii) We proceed as follows. Choose R and D* cS2(R) as in (i) and notice that

the corresponding curvature is given by

K o =27z/A.

From (3.9), one obtains,

G*'(r) A'(r)=47zA(r) (1 - A ( r ) t

2A]"

From (3.5) and the isoperimetric inequality in $2(1), we have that

G'(r)

A'(r)> (L(r)) z >__

4xA(r) (1

47r ]' It follows from the above that

Since 27;_<A_<47z, we can write A=(1 +e)2zc, 0<~_< 1. Thus 47z-2A/1 +~ and, since A(r) < A,

It follows that

l _ A ( r ) = 1 A ( r ) ( l + 8 ) > 1 A(r)

4rc 2A 2A 2"

4~ ] - 2A / < G'(r) ).

Thus, bydividing (1-A4(~r)),usingthatA(r)<A, andintegratingonr,

4re

Since H = H * , e = ( A - 2rc)/2zc, and D* is a hemisphere of

S2(R),

we obtain finally,

4~r 2re

- 2 = ;~ (D*)_-< ;~ (D) a~ - A'

hence

21(D)>_ 2(4re-A). q.e.d.

A

(3.11)

Remark.

Proposition (3.10) holds even ifD is not simply-connected. This

follows from the fact that we have used the isoperimetric inequality only for domains in a sphere where it holds independently of simple-connectedness.

(3.12) Sometimes it is convenient to have Proposition (3.3) in a form that can be applied to certain degenerate situations. The following is a useful example.

Call a metric

ds 2

in

M degenerate

if

ds 2

is allowed to be zero at isolated

points (to be called

singular points

of the metric). For such metrics, the

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 21

(3.13) Proposition. Let D c M be simply-connected and denote by d s 2 the metric of M. Let F be a non-negative C2-function on M which vanishes only at isolated points, and denote by 2~ the first eigenvalue of the problem

(3.14) A u + 2Fu=O, u~H(D).

Consider M with the generalized metric d~r 2 = F ds z and assume that the Gaussian curvature Is of (M, dcr 2) satisfies I( <=K o. Then

21 > 21(D*),

where D * ~ M ( K o ) is a geodesic disk with the same area as the area of D in (m, da2).

Proof We will denote the elements in the degenerate metric by a hat; thus d2~ is the area element, and A is the Laplacian in (M, do-Z). Away from the singular points, the following relations are easily verified:

A = ( 1 / r ) A , d ~ l = F d m ; and if u is a C2-function on M,

Ir Vull 2 = (l/V)[I

Vul[ 2.

We will first prove that, in the generalized metric,

[I ~ull 2 dI~l

(3.15)

; _ o

S u ~ d M '

)3

where u is a solution of (3.14). To prove (3.15), we consider non-overlapping disks D~(pi) of radii e > 0 around the singular points P l .... , Pk of dcr z in/). Set D~

k

= U D~(p~), denote by ds the element of arc of 0D~ and let n be a unit normal i=1

vector along the boundary ODe of D~. Then, from (3.14), we obtain

S

D - D~

u 2 d ) ~ l = ~ ) ~ l ' u 2 d M = - S u A u d M

D-D~ D-D~

= (. IlVull2dM+ [. u<Vu, n>ds

D-- De OD~

= ~ I[~u[12d~+ 89 ~

(Vu2, n)ds.

D D~ D--D~

Since u 2 is differentiable,

as we claimed.

lira y

(Vu 2, 11)

ds=O. Thus ~ 0 OD~

,~ .[

u a d ~ = y II PujI2 d ~ ,

22 J.L. Barbosa and M. do Carmo

Now, let u be a first eigenfunction of (3.14). Then u > 0 , and u(p)=0 if and only ifpEOD. Then, as in the proof of Proposition (3.3), we can use u to define a function u* on the "symmetrized" domain D*< M(Ko) and obtain that

G(0) > G* (0) > 2, (D). q.e.d. 21 = H ( 0 ) = H , ( 0 )

The various forms of Proposition (3.3) depend on an isoperimetric inequality for surfaces with bounded curvature. Actually such an isoperimetric inequality can be extended to finitely-connected domains contained in simply-connected domains provided that condition (3.17) below is satisfied.

(3.16) Proposition. Let M be a C2-differentiable manifold endowed with a C 2- generalized-Riemannian metric and assume that K <=K o. Assume that D c M is a finitely-connected domain contained in a simply-connected domain D s and set A s

= area Ds. Assume further that

(3.17) KoA~<= 4rr.

Set A = area D and L = length •D. Then

( KoA

(3.18) L2>4~zA 1 - ~ - ] ,

equality holds if and only if D is a geodesic disk in M(Ko).

Proof. Let D o be the smallest simply-connected domain that contains D. It is easily seen that D is obtained from D o by deleting a finite number of closed disks b I . . . /5 k. Let us first consider the case where D = D o - D 1. Set L 1 =length 0D 1, L0=length0D0, A l = a r e a D 1, A 2 = a r e a D a. Then A = A o - A 1, L = L o + L 1. Since (3.18) applies to both D o and D 1 (see [2]), we obtain

L 2 + L 2 => 4n(A 0 +A~)-Ko(A2o +A2). Thus

( L o + L 1 ) 2 - 4 r c ( A o - A 1 ) (1 K ~ 1 7 6 = + 2 A I ( 4 r c - K ~ 1 7 6 o r

(3.19) L 2 - 4 ~ A ( 1 K ~ 1 7 6 =

Since K o A o < K o A ~ < 4 n , we obtain the desired inequality. Equality implies that L o L 1 +Al(47c-KoAo)=O, which implies that L 1 =0. Thus D = D o, and we can use the equality case of (3.18) to complete the proof in this special case.

The above argument applies whenever D can be written as a difference of two sets and (3.18) applies to both of them. Therefore we can use an inductive procedure to conclude the proof of Proposition (3.16). q.e.d.

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 23

(3.21) Remark. Inequality (3.19) shows that if a doubly-connected domain D of a surface M with nonpositive curvature is contained in a simply-connected domain, then

(3.22) I~ - 4 h A > 2 L 1L 2 .

where A = a r e a D , L~ and L 2 a r e the lengths of the individual boundary curves and L = L 1 + L 2. This is not true if D is not contained in a simply-connected domain, as a simple example shows, and is related to a result of Osserman [14].

4. Proofs of the Theorems 1.2, 1.3 and 1.4

(4.1) Let x: M--*A3(a) be a minimal immersion, where A3(a) is either S3(a) or H3(a). Let D ~ M and let V be a normal vector field o n / 5 that vanishes on ~D. Choose an adapted frame e 1, e2, e3; then V = u e 3, where u is a function that vanishes on 0D, and the formula for the second variation along V is (E73, p. 47)

I(V, V ) = S ( - u A u - ( ~ R i 3 i 3 ) u Z - ( ~ . . h ~ ) u Z ) d M , i , j = 1,2,

/ ) i zj

where R~3 j3 are components of the curvature tensor of A 3 (a) in the frame el, e z,

e 3 and h~j are the components of the second fundamental form of x in el, e 2. By setting, as usual, ~ h ~ = IIBII 2 and noticing that Ri3 ~3=a, we obtain

ij

I(V, V) = ~ ( - uA u - 2 a u 2 - lIB II z u 2) d M .

D

By Gauss' formula, - I I B [ l a = 2 K - 2 a , where K is the curvature of the metric induced on M by x. Thus, we obtain finally

(4.2)

I(v, v)= S

( - u A u + 2 ( K - 2 a ) u 2) aM. D

(4.3) P r o o f o f Theorem 1.2. Assume that D is not stable, that is, there exists a normal vector field V o n / 5 , vanishing on c~D, such that I(V, V ) < 0 . By Smale's version of the Morse index theorem ([193 and [17]), there exists a domain

D ' ~ D and a Jacobi field J = u e 3 on D' that vanishes on ~D'. This means that u satisfies:

(4.4) - A u + 2 u ( K - 2 a ) = O in D', u = 0 on ~3D'.

Notice that, by (4.4), if u ( p ) = 0 for p~D', then u changes sign in D' ([11], p. 210). Thus we can assume that D' is so chosen that u in not zero in D'. Since K

- 2 a < 0 we can introduce in M the metric

do -2 = ( 2 a - K ) ds 2,

w h e r e d s 2 is the metric induced by x: M ~ S 3 ( a ) . In the metric d o "z, Equa-

tion (4.4) becomes

24 J.L. Barbosa and M. do Carmo

where, as usual, we denote by a hat the corresponding elements of the metric

do -2. Equation (4.5) means that the first eigenvalue 2 I(D') = 2. By Proposition 2.2,

the Gaussian curvature /( < 1. By Proposition 3.3 and Corollary 3.20, %~(D')>2~(D*), where D* is a geodesic disk in a sphere $2(1) with curvature 1 and area of D* is equal to the area ~(D') of D' in the metric do -2. But

A(D')= S (2a-K)dM<Zzc.

D'

It follows that D* is contained in a hemisphere of $2(1) the first eigenvalue of which is 2. Thus

2=s

what is a contradiction, and proves Theorem 1.2 modulo the statement on the sharpness of the bound 2re.

To show that the bound is sharp, we need the following Lemma.

(4.6) Lemma. Let x: M---,S3(a) be a minimal immersion. Let D c M be a simply- connected domain such that 2 I(D) =)~1 < 2a, where 21(D ) is the first eigenvalue of D in the induced metric. Then D is unstable.

Proof Choose an eigenfunction u on D, corresponding to the eigenvalue 2 I. Thus

(4.7) Au+21u=O , u = 0 on #D.

Set V=ue 3. By (4.2) and the fact that K < a , we obtain

I(V, V)= ~ ( - u A u + 2 ( K - Z a ) u Z ) d M < ~ ( - u A u - ZauZ) dM.

D D

By Stokes' theorem and the fact that u = 0 on 0D,

- j u A u d M = ~. IlVullZdM=21 ~ u2dM.

D D D

Thus

I(V, V)<()~ 1-2a) S u2dM <O,

hence the area of D decreases in a variation given by V, and D is unstable, q.e.d.

(4.7) Example. Let x: S2(a)--~S3(a) be the inclusion of S2(a) as an equator of S3(a); x is clearly minimal. Let e > 0 be given and set c~=1+8. Let D~ be a

2u

geodesic disk in S2(a) with area - - c~. Then D~ contains a hemisphere H of S2(a).

a

Since 2 I(H) = 2a, we have that 21(D~)< 2a, and, by Lemma (4.6), D~ is unstable. On the other hand, since K - a ,

( 2 a - K ) d M = a ~ d M = Z u ( l + e ) .

D~ D~

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 25 Example (4.7) shows that the bound 2n of Theorem (1.2) is sharp and this concludes the proof of Theorem (1.2). q.e.d.

(4.8) Proof of Theorem (1.3). We first observe that, since a < 0 ,

ID(V,

V ) = S

(-uAu+ 2ua(K--2a))dM

D

>= ~ ( - u A u + 2 u a K ) dM=iD(V, V),

D

for any normal vector V = u e 3 that vanishes in 0D (here we have used 11) to denote the index form in the domain D). To show that D is stable, it suffices to show that [D(V, V ) > 0 for all such V.

The proof now is similar to the proof of Theorem (1.2). Assume that D is not stable. Then there exists a V such that [D(V, V)=<0. By Smale's theorem [19], there is a domain D ' c D and a function u " / 5 ' ~ R that is positive in D', vanishes on 0D' and, in D', satisfies

(4.9) - A u + 2Ku=O.

By the conformal transformation do "2= - K d s 2 we change (4.9) into (4.10) A u + 2 u = O , u = 0 on 0D', u > 0 on D'.

By using Propositions (2.2) and (3.3) we show, in the same way as in the proof of Theorem(1.2), that (4.10) leads to a contradiction. This proves Theorem (1.3). q.e.d.

(4.11) Before proving Theorem (1.4) let us fix some notation. Let x: M ~ R " be a minimal immersion. Choose an adapted frame e 1 .... , e, in R"; thus el, e z are tangent to x(M) and e3, ..., e, are normal vectors. Let us agree in the following range of indices:

i , j , k = l , 2 , c~, fi, 7 = 3 , ..., n , A , B , C = I , . . . , n .

Let D c M be a domain and denote by V = ~ V~e~ a normal vector field that

vanishes on #D. Then the formula for the second variation reads

(4.12) I(V, V)= ~ {~ ( - V~A V~, - ~ h,~jh,~: V B V~)} dM,

a i j B

where h~j are the coefficients of the second fundamental form in the direction of e~, and A V~ is the c~-component of the Laplacian of V (see [7], p. 46).

V is a Jacobi field if

AV~+ ~ hi~jhi~Vt~=O.

(4.13)

26 J.L. B a r b o s a a n d M. do C a r m o

(4.14)

Proof of Theorem

(1.4). Assume that D is not stable. Then by Smale's theorem, there exists a domain

D ' c D

and a Jacobi field V o n / 5 ' vanishing on ~?D'. We can assuem that the only zeroes of V in D r are isolated points ql . . . qm" Away from these points, choose an adapted frame e 1 .... , e, so that

V=ue 3.

With such a choice, the e3-component of (4.13) becomes

0 = ( A ( u e 3 ) , e 3 ) 4- ( 2 h{3j) u = A u + u ( ( A e3, e 3 ) 4- E h23j)

ij ij

= A u - u ( ~ ,

Ic03,L2-~ Ic%il2),

c~ i 2

where Io)3AI - ~ (O3A(ej)) 2 and coa~ are the pull-back of the connection forms of

J

R" by the immersion x. By setting

A2=21c03~12+ ~ Ic%12,

c~ i , ~ > 3

and noticing that, by Gauss equation, - 2 K = ~ 09 2 I i~l , w e obtain

-21 3,1

y,

2

c~ i ~ i i , ~ > 3 i , ~ > 3

= 2 K + A 2.

ItfollowsthatuispositiveinD'-{~)i=t qi}=WvanishesinOWandsatisfies

(4.15)

A u - 2 K u - A 2 u = O

in W.

Furthermore, u 2 is analytic in all of D'.

Now consider the first eigenvalue 21 for the problem:

(4.16)

A f + A ( - K ) f = O

in D', f = 0 in ~?D'.

We claim that (4.15) implies that 21 <2. The proof is similar to the first part of the proof of Proposition (3.13). We cover the points q~ by small disks

D~(q~),

set

O~= ~ O~(ql ),

and obtain from (4.15)

t

2 ~ (-K)u2dM>= -

~ uAudM

D'-D~ D'--D~

= ~ IlVull2dM+ 89 ~ (Vu2, n)ds,

D' -- D~ ODe

hence, since u 2 is analytic,

[IVull2 dM

2>--

D' ~'~1'

-- ~ ( - K ) u 2dM

D'

as we claimed.

Stability of Minimal Surfaces and Eigenvalues of the Laplacian 27

/~ of d 6 2 satisfies /s < 2. Notice that D' is finitely-connected, contained in a simply-connected domain, and satisfies condition (3.17). Thus we can apply Proposition(3.13) and Corollary(3.20) to conclude that 21>2~(D*), where 21(D* ) is the first eigenvalue of the Laplacian of a geodesic disk D* in a sphere

S (~22) of curvature equal to 2 such that the area of D* is the area of D' in the

metric d o "z , i.e.,

area (D*) = ~ dM = ~ ( - K) dM < ~ [K[ dM < (4/3) ~z.

D' D' D

On the other hand, it can be easily computed by Proposition 3.10(ii) that for

such a domain D*~S2 ( ~ 2 ) , we have )Ll(D*)> 2. Thus

(4.17) 2>)~ 1 > 2 z ( D * ) > 2 ,

what is a contradiction and completes the proof of Theorem (1.4).

(4.18) Remark. When n = 3 , the method of proof of Theorem(1.4) applies to show that if ~ ]K] d M < 2 n then D is stable (this is a weaker form of the result

D

proved in [1]). We have only to remark that for n = 3 , /s Thus D* is properly contained in a hemisphere of $2(1), whose first eigenvalue is 2, and we arrive at the same contradiction as in (4.17).

5. Final R e m a r k s

(5.1) If one looks for a better bound to replace (4/3)~ in Theorem (1.4), such a bound would necessarily be smaller than or equal to 2~c. To see this for n > 5 , notice that we have in the Enneper's surface f : R 2 - , R 3 a simply-connected domain D ~ R 3 with the following properties: ~ ]K]dM=2rc+e 2, and D~ is

o

unstable, i.e., there exists a variation that decreases area. Let g: D ~ R m, m > 2 be an almost flat minimal immersion of D~ (if m = 2, we take g to be flat). Then, for t small, the product immersion f + tg: D ~ R m+3 is unstable. By letting ~ ~ 0, one obtains examples of minimal surfaces in R", n > 5, the absolute total curvature of which are arbitrarily close to 2n.

For the case n = 4 , one can proceed as follows. Let A: 112 x R ~ S O ( 4 , C) be a map that is holomorphic in the first argument, differentiable in the second, and satisfies A(z, 0)=Id., for all zOl~; the group S0(4, II;) acts on the hyperquadric Qz~P3(II~). Now, let ~b be the Gauss map of the Enneper surface f, and define a family of minimal immersions in R 4 by x ~ ( z ) = R e ~ A 4 ( z , t ) d z . For t small,

xt: D~--,R 4 is unstable, and, by letting e ~ 0 , its total curvature can be made as close to 2n as we wish.

28 J.L. Barbosa and M. do Carmo

m i n i m a l i m m e r s i o n x: M ~ R n s a t i s f i e s / ~ _< a _< 2, then the c o n d i t i o n

4 ~

S

I / I

dM<(a+l

~

D

implies the stability of D c M . In particular, i f / ~ < 1 and ~

IKI <2~z,

D is stable. D

(5.3) Chen has shown [6] that if x:

M~R", n>4,

is a complete minimal immersion of a surface M with total curvature 2n, then M is conformal to the complex plane and n = 4 . It follows from his proof that, by choosing an appropriate basis for R 4 = C • II~, x can be represented in the form x: I1~ x ~72,

x(z)--(z, az2+bz+c).

Thus by Wirtinger's inequality ([-10], p. 34), x is globally stable.

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Studies 21. Princeton: Princeton University Press 1951

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