Stability of minimal surfaces in a 3dimensional hyperbolic space

Arch. Math., Vol. 36, 554--557 ( 1 9 8 1 ) 0003-889X/81/3606-0004 $ 01.50 ~- 0.20/0 9 1981 Birkhi~user Verlag, Basel

Stability of minimal surfaces in a 3-dimensional hyperbolic

space

By

J. L. B~Bosx and M. DO CA_aMO

1. Introduction.

(1.1). L e t x: M n --~ Mn+l be a minimal immersion of a n-dimensional orientable manifold into a n ~- 1-dimensional Riemannian manifold _~]. Let D be a domain in M with c o m p a c t closure D and pieeewise s m o o t h boundary 8D. D is then a critical point for the area function of the induced metric, for all variations of J0 keeping aD fixed. We say t h a t D is stable when such a critical point is a relative minimum. The geometrical characterization of stable domains is the main problem on the subject. F o r references on this we mention [1] and [3].

I n this p a p e r we prove the following theorem t h a t is an improvement of Theor. (1.3) of [1]. We do not know whether the result is sharp. L e t K denote the Gaussian curvature of M in the induced metric.

(1.2) Theorem. Let x: M 2 --> Ha(b) be an isometric immersion o/ M into the hyper- bolic space H 3 (b) with constant curvature b < O. I f D c M is simply connected and

f ([K I -}- - } b ) d i < 2•

then D is stable.

2. General observations.

(2.1). The question whether D is stable or not is naturally studied b y looking a t the sign of the second variation formula. This is a quadratic form 11) t h a t acts on normal vector fields to M t h a t vanish on aD. Simons [5] has given an expression . for ID in a quite general set up. I n the case of eodimension one, b y choosing a unit normal vector field N, 11) can be t h o u g h t as an operator t h a t acts in the space of C ~~ real functions on D vanishing on aD, given b y

(2.2) ~ ( - - u A u -- (_~ -~ U A H2)u2)dM, D

Vol. 36, 1981 Stability of minimal surfaces 555 tion of the equation

(2.3) A u + (R -+- II A [I2)u -= 0

in U is called a Jacobi/ield in U. I t is known t h a t D is stable if and only if there is no Jacobi defined in U c J0 t h a t is zero on 0U. L e t hi(D) be the first eigenvalu e of the Laplacian on D.

(2.4) Proposition. Let x: M n -~ _~n+i be a minimal immersion. Assume D r M is a domain with smooth boundary and that the sectional curvatures Ra of ~ satis/y bi ~ Ra ~ b2. Now we have:

(a) i/ ]1 A ti 2 + nbl > J~i

(D)

then D is unstable. (b) i/ iI A[I 2 + nb2 < 2i (D) then D is stable.

(2.5). As a consequence of this proposition it follows t h a t if b2 ~ 0 and ]I A t[ 2 - - n b 2 t h e n M is globally stable. I n particular, if b2 ~ 0 a n y totally geodesic sub- manifold is globally stable. I t also follows t h a t as much as b2 becomes negative as much as one can expect to find non trivial examples of minimal submanifolds of t h a t are globally stable.

(2.6). The proof of the above proposition follows from the observation t h a t if bl ~ R~ ~ b2 then nbi ~ ~ ~ nb2 and

--.[uAudM

> 2l(D) S u 2 d M

D D

for all pieeewise smooth u t h a t are zero on aD. Equality occurs if and only if u is a first eigenfunetion for zJ.

3. A curvature estimate.

(3.1). Let x: M 2 --> •3 be a minimal immersion. Denote b y ds 2 the metric on M induced b y x and b y A its second fundamental form. L e t ei, e2, e3 be an adapted frame to the immersion x, let wi, w2 be the dual eoframe to el, e2 and let hij, i, ?" = 1, 2, be the coefficients of the second fundamental form of x in the frame el, e2. Then

Ii A =

Let I~ABCD , A , B, C, D = 1,

2, 3,

be the coefficients of the curvature form o f / ~ . I f a is the plane generated b y eA, eB then R~ =

RABAB.

Set u ---- 89 H A I[ 2 + a, a > O, and da 2 = uds 2. We are interested in finding an upper bound for the Gauss curva- t u r e / ~ of (M, da2).

(3.2) Lemma. Let .M have constant curvature c and let a ~ m a x (c, --c/3), a > 0.

Then ~ <__

1.

(3.3) P r o o f . I t is easily computed t h a t (see e.g. [2] eq. (2.8)) the Gaussian curva- t u r e / ~ of da 2 = uds2 is given b y

K

(3.4)

_l~"

= __ - } - _ _ - - Au -}- _ _ . )

556 J.L. BARBOSA and M. DO CARMO ARCH. MATH. where K is the Gaussian c u r v a t u r e of de 2, A u is the L a p l a c i a n of u in the metric ds2 a n d u l , u2 are defined b y

(3.5)

d u = u l w l + u s w 2 .

Since u = 89 s + a = 89 Z h ~ + a, we obtain

T h e gauss equation states t h a t (3.7) R1212 - - K = 89 ][ A ]l s .

S u b s t i t u t i o n o f (3.6) and (3.7) in (3.4) yields

(3.8) g = --u (R1213 + a - - u) + ~ (~hi/h0.k) 2 _ 21 ~hrshi: ~

a 1

Since ~ has constant c u r v a t u r e , following [4] a n d using t h a t 1 ~ i, ] ~ 2, one ob- tains

Since RABAB --- C, we t h e n h a v e

~ h ~ j A]~i = 2 c I[ A I[ 2 -- ]I S []4.

F u r t h e r m o r e , one can easily check (see [1], proof of Prop. (2.2)) t h a t the second t e r m on t h e right h a n d side of (3.8) is zero. I t follows t h a t

1 a 1

(3.9) /~ = --u (c ~- a - - u) - - ~ - ~ u a ~hi~ k - - ~-u2 (4 c (u - - a) - - 4 (u - - a) 2) a n d b y dropping the second t e r m on the right h a n d side of (3.9), we obtain

g __< l(u)

where

1

] (u) = 1 ~- ~ (2 a (a -~ c) - - (3 a -~ c ) u ) .

N o w we need t o p r o v e t h a t , for u --> a _> m a x ( c , - - c / 3 ) , we h a v e / ( u ) < 1. T o do this, one studies separately t h e case c > 0 and c < 0 a n d shows t h a t t h e gTaphs of ] for u ~ a are below 1 a n d a p p r o a c h a s y m p t o t i c a l l y 1 as u --> oo. This p r o v e s L e m m a (3.2).

4. P r o o f of T h e o r e m ( 1 . 2 ) .

Vol. 36, 1981 Stability of minimal surfaces 557

restricted to the b o u n d a r y of D is zero, t h e n

(4.2) Iv(u) = ~ (-- u A u -- (TO § [I A l[2)u2)dM ,

D

where now ~ = 2 b < 0. B y the conformal change of metric:

(4.3)

d(~2 =

( 89 It A 112

+ a) d s 2 ,

where a ~ m a x (b, --b/3), a n d a > 0, we o b t a i n

( 2 b § )

(4.4) ID(U)--~ ] - - u A a u - - 2 u s dMa

2 a § ]IA [[2 '

where dMa a n d Aa are respectively t h e area element and the Laplaeian o p e r a t o r

in the metric d(~ 2. I t follows t h a t

(4.5) ID(U) ~ f ( - - u A a u -- 2u~)dMa ~ (~1 -- 2) ]u~dMa,

D

where 21 is the first eigenvalue o f Aa on D. According t o L e m m a (3.2), t h e Gaussian c u r v a t u r e K a of the metric d a 2 satisfies K a ~ 1. F r o m P r o p o s i t i o n (3.3) o f [1] we have t h a t

(4.6) 21 > ~1 (D*),

where D * is a geodesic disk on the sphere S 2 (1) such t h a t a r e a a D = area D*. B u t , b y the Gauss equation,

(4.7) a r e a D * = S ( 8 9 2 4 7 ~([KI §

Hence D * is contained in a hemisphere a n d so 21(D*) > 2. Therefore ID(u) > 0

and D is stable. This completes t h e p r o o f of T h e o r e m (1.2).

R e l e r e n e e s

[1] J. L. B.~RI3OS~. and M. Do CARMO, Stability of minimal surfaces and eigenvalues of the Lapla- clan. To appear at the Math. Z.

[2] J. L. BA-~BOSA and M. :DO CAI~O, A necessary condition for a metric to be isometrically and minimally immersed in/~n+l. An. Aead. Brasil. Cienc. 50, 415--454 (1978).

[3] M. Do C~_RMO, Stability of minimal submanifolds. To appear in LN-~.

[4] S. S. C~F, RI~, Minimal Submanifolds in a Riemannian manifold. University of Kansas, Dept. of Math., Tech. Rep. 19, Kansas 1968.

[Sj J. SIMO~S, Minimal Varieties in Riemannian Manifolds. Ann. Math. 88, 62--105 (1968).

Anschrift der Autoren: g. L. Barbosa

Universidade Federal do Cear~ Instituto de Matem~tica Fortaleza, Cear~, Brazil

Eingegangen am7.8.1980

M. do Carmo IMPA