Minimal Surfaces and H-Surfaces in Nonpositively Curved Space Forms

Minimal Surfaces and H-Surfaces in Nonpositively Curved Space FormsAuthor(s): Bennett PalmerSource: Proceedings of the American Mathematical Society, Vol. 119, No. 1 (Sep., 1993), pp.245-250Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2159848 .

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 119, Number 1, September 1993

MINIMAL SURFACES AND H-SURFACES IN NONPOSITIVELY CURVED SPACE FORMS

BENNETT PALMER

(Communicated by Peter Li)

ABSTRACT. We show that if the Gauss curvature of a surface of constant mean curvature in a nonpositively curved space form is sufficiently pinched, the surface is stable. In this case, we also give an upper bound for the inradius. We then show that the inradius of a stable minimal surface in Euclidean space, which is contained in a solid cylinder, is bounded above by a constant depending only on the radius of the cylinder.

Let M3(c) denote a 3-dimensional oriented space form of constant sectional curvature c < 0. Let X: M -_ M3(c) be a smooth immersion of a smoothly bounded surface M with curvature K and mean curvature h. Set

K = max K, K = min K. M M

We show

Theorem I. For h, c E R with -A2 := h2+c < 0O there exist universal constants w(c, h) > e2 with the following property:

If M c M3(c) is a smooth orientable surface with constant mean curvature h and

(-A2 _ K)/(-A2 - K) < w(c, h)

then M is stable. In addition, 7.4 ... = e2 < ?((0, 0) < 10.75 ... holds.

Theorem II. If -o0 < K, K < -A2, and M contains a geodesic ball Br(xo) of radius r then

r2 < ______K)__1ge[(-A - K)/(-A2 K)].

In the second part of this paper we consider surfaces in EN which are extrinsically bounded in some way. Let CR = {X = (X1, X2, X3) E E3 I x2 + x2 <

R2}. We show

Theorem III. There exists a constant cl > 0 with the following property: If M c E3 is an orientable stable minimal surface with Br(p) c M and M C CRI \CR2 for some R1 > R2 > 0 then r2 < cl (R2 - R2).

Received by the editors September 11, 1991. 1991 Mathematics Subject Classification. Primary 53A 10.

? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

245



246 BENNETT PALMER

Before beginning the proofs, we review some basic facts about surfaces in Ml (c).

Let M -, M3 (c) be an orientable surface. Let ds2 denote the induced metric. (M, ds2) may be considered a Riemann surface in a natural way by introducing isothermal coordinates (x, y) and using z = x + iy as a com- plex coordinate. Doing so, ds2 may be expressed as ds2 = ePldzI2 and the curvature is

(1) K = -2e-Ppz2.

The second fundamental form of M has an expression

El = Re{qddz2 + hePdz dz}

where Odz2 is an invariant quadratic differential and h is the mean curvature. The fundamental equations of the immersion are those of Gauss

(2) 1bI2 = e2p(h2+ c - K)

and Codazzi

(3) = ePhz.

When h = const, (3) implies that b is holomorphic in z. It then follows that either 0 0 and M is totally umbilic or the zeros of X are isolated.

Lemma 1. Let M c M3(c), c < 0, have constant mean curvature h such that h2 + c < 0. Assume M has no umbilics. Then the conformal metric

(4) dS2 (h2 + c - K)ds2

has curvature K satisfying

(5) k> 1. Proof. Using (1)-(3) one has

(6) 0 = Alogkbl =Alog(-A2- K)1/2 +Ap

= Alog(-A2 - K)1/2 - 2K

where A = 4e-Pazaz. Therefore, using (1) to compute K one has

K = -(-A2 - K)-'{Alog(-A2 - K)1/2 - K} = (_A2 - K) K).

Since K < 0 on M, (5) follows.

Proposition 1. Again assume h = const, h2 + c =-A2 < 0 and

(7) 0 < -A2 -K < -A2 -K < -A 2- K < oo.

Then the first (nontrivial) eigenvalue AI of the problem

A+2A(-A 2 -K)V=O onM,

( 2 A=K on aM

satisfies

(8) [I log((-A2 - K)/(-A2 - K))]-j < l



MINIMAL SURFACES AND H-SURFACES 247

Proof. Let ,V > 0 be the eigenfunction corresponding to 2A . Let g(x, y) denote the positive Green's function of M. Then

/(X) = Al (-2A2 - 2K(y)) V(y)g(x, y) * 1 (y)

< 2Aj -2K(y)yi(y)g(x, y) * 1(y).

Therefore,

I V(Y)I ? <i 11 V/100 J-2K(y)g(x, y) * 1(y) -i AlIVllv(x),

where v solves

(9) Av = 2K in M, v = O on AM.

Choosing x where v achieves its maximum we arrive at

(10) 1 ? l1I~II.

By (6) and (9) A(v - log(-A2 - K)1/2) - 0 in M

and on dM v - log(-A2 - K)1/2 < - log(-A2 K)1/2.

It follows from the maximum principle and (8) that on M

( 1 ) v < log(-A2 - K) 1/2 - log(-A2 - K)1/2

< log(-A2 - K) 1/2 - log(-A2 - K)1/2.

Using this and (10), (8) follows.

Proof of Theorem I. The surface M is stationary for the functional

J = area + 2H(enclosed 3-volume).

The second variation of J for variations of the form V - N where N is the unit normal to M and q e CO (M) is given by 52J = f-VL. Here L is the selfadjoint elliptic operator LyI = AV + 2(-2A2 - K) V. Assuming the hypothesis of the theorem, we have by Proposition 1, Al >? 1. Using integration by parts we obtain

52J = J-VLV = J(IVyVI2 - 2(-2A2 - K) V2)

? J(JVVI2 - 2(-A2 - K)yV2) > 0

and M is stable.

The upper bound for wj(0, 0) follows from Example I.

Remark. We have shown that under the hypothesis of Theorem I, the second variation of J is nonnegative for all compactly supported variations. When h 54 0 this is stronger than the condition that 52J be nonnegative for all volume-preserving variations (cf. [B-DC]).



248 BENNETT PALMER

Example I. Let C C 1R3 C x R be the catenoid parameterized by X(u, v) = (ely cosh(u), u), (u, v) E IR x [0, 27r). One easily computes that the curvature is given by K = -(cosh u)-4 and that the support function is given by s = -1 + u tanh(u) .

Denote by Qt the symmetric "waist" domain of the catenoid given by ulu < t. Then Qt will be stable as long as s is negative, that is, for t < ul 1.2... For Ut

e2 > K/K = (cosh t)4

holds for t < t1 1.0850.... It follows that Theorem I has correctly predicted stability in this case. Furthermore, for Qu1 , K/K 10.75... furnishes the upper bound in the corollary. Finally the total curvature of Qt is

rti

27r cosh-2 udu 27r(1.590 ... ) > 27r. -tj

Consequently the criteria of Theorem I is independent of the Barbosa-DoCarmo result [B-DC 1].

To prove Theorem II we state without proof a special case of an eigenvalue estimate due to Gage [G].

Theorem (Gage). Let Br be a geodesic ball of radius P contained in a surface of curvature K > -fl2 = const. Then the first Dirichlet eigenvalue of the Laplacian A on Br satisfies

(12) il< 7r2/l2 + 32/4.

Proof of Theorem II. For a region Q c M let 24 (Q) denote the first Dirichlet eigenvalue of A for U. Here A is the Laplacian for the metric d9 of Lemma 1. Let AI (Q) be the first eigenvalue of the problem

(13) y/A + 2A(-A2 - K)y = 0 in Q,

V = O on aQ.

Since A = (-A2 - K)-lA, it is clear that AlI(Q) = 2A1 (Q). Let y be a mini- mizing geodesic of length P for the metric dg. Then

P J(-A2 - K)1/2 ds > (-A2 _ K)1/2 ds.

It follows that Br( A2 K)/2 C Br. By a well-known monotonicity property of eigenvalues

A I(Br(-A2-K)I1/2) > A I(Br) = 2A1 (Br)

So by Lemma 1 and Gage's Theorem with /3 = 0, we have

r2(-A2 - K) - 2Ai(Br).

Combining this with the lower bound of Proposition 1 yields the result.

We now consider surfaces in EN which are extrinsically bounded.



MINIMAL SURFACES AND H-SURFACES 249

Lemma 2. Let M C E3 be a minimal surface. Let Q c M be a smoothly bounded subdomain, and let /LI be the first Dirichlet eigenvalue of the Laplacian in Q. Assume

(14) MCCRI\CR2 R1 >R2>O.

Then

(15) 2i > 2 Il1?RR2'

1 2

Proof. Define X = (X2 +X2) . Let N = (Ni, N2, N3) be a unit normal defined on a neighborhood in M. Then

AT = 2 E (2xiAxi + 2IIVxi|j2) = jjVx1j2 + j1Vx2112 2i=1 ,2

- -N + 1 - N= 1+N? 1.

By (14) we have

(16) 2 < < 21_ Let O > 0 be a solution of AV +,ug=O with Vg O on aQ. Then

Vg(X) = p1i j Ig(y)g(x, y) * 1(y)

and consequently

qI(x)j <pil~l KIojg(xy)* 1(y).

Taking x values where V/ achieves its maximum yields

(17) <-I mAtxe Jg(x, y) * 1(y) - #I max S(x)

where S solves AS=-l inKQ,

S-O on a&. Therefore,

A(S+ T)>O in Q and S + T = T < R2/2 on aQ.

By the maximum principle

S < _ T < 1 2 in Q. - 2 - 2 2 Combining this with (17) proves (15).

Proof of Theorem III. Since M is stable, it follows by a result of Schoen [Sc, Corollary 4] that there is an estimate K(x) > -2a/r2, for all x E Br12, where a is a universal constant. Using this lower bound for K in Gage's upper bound for juI gives

a R2 4C1 U1 <4r2+ r2 = 2 2

Combining this with the lower bound for 4u1 in Lemma 2 gives the result.



250 BENNETT PALMER

Corollary. Let M C E3 be a complete minimal surface. If there exists p E M such that

(18) lim sup (-K) = 0 r--oot r B(P)

then M is not contained in a cylinder. Proof. Assume to the contrary that M c CR for some R, 0 < R < oc. Let ro be a constant with r0 > cl R2 with cl as in Theorem III. Then any disc Bro c M is unstable. By a result of Barbosa and DoCarmo [B-DC1] fB (-K) > 27r.

Taking the sequence r, = nro, one finds that since Brn(p) contains at least n disjoint geodesic balls of radius ro,

1 [~~K~ 27rn 27i (-K) >-~~= _ >> 0~ rn JB(P) nro rO

giving a contradiction.

ACKNOWLEDGMENT

We would like to thank Professor Robert Osserman for his helpful comments.

REFERENCES

[B-DC] J. L. Barbosa and M. DoCarmo, Stability of hypersurfaces with constant mean curvatures, Math. Z. 185 (1984), 339-353.

[B-DC 1] , On the size of a stable minimal surface in R3, Amer. J. Math. 98 (1976), 5 15-528.

[F-C-S] D. Fischer-Colbrie and R. Schoen, The structure of complete stable, minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211.

[G] M. Gage, Upper bounds for the first Dirichlet eigenvalue of the Laplace-Beltrami operators, Indiana Univ. Math. J. 29 (1980), 897-912.

[0] R. Osserman, Global properties of minimal surfaces in E3 and En , Ann. of Math. (2) 80 (1964), 340-364.

[S] M. Spivak, A comprehensive introduction to differential geometry, Vol. IV, Publish or Perish, Boston, MA, 1975.

[Sc] R. Schoen, Estimates for stable minimal surfaces in three dimensional manifolds, Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, NJ, 1983.

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