Comment. Math. Helvetici 64 (1989) 3 4 - 4 3 0010-2571/89/010034-10501.50 + 0.20/0 1989 Birkh~iuser Verlag, Basel

Complete minimal surfaces with index one and stable constant mean curvature surfaces

FRANCISCO J. LOPEZ and ANTONIO ROS

Introduction

Minimal surfaces and surfaces with constant mean curvature in the Euclidean space R 3 a r e critical points of variational problems. In the first case the surfaces have critical area for all variations with compact support. In the second one their area is critical for compactly supported variations that leave constant the signed volume "enclosed" by the surface.

So problems on stability and index of their Jacobi operator arises naturally. Do Carmo, Peng [2], Fischer-Colbrie, Schoen [4], and Pogorelov [11}, have proved independently that the plane is the only stable complete minimal surface in R 3. Quite recently Fischer-Colbrie [3], and Gulliver, Lawson [6], [5] showed that a complete orientable minimal surface has finite index (in the sense that the index of the Jacobi operator on every bounded domain has a uniform upper bound) if and only if it has finite total curvature. The study of the stability for surfaces with constant mean curvature was made, in the compact case, by Barbosa and do Carmo [1]. They proved that the sphere is the only compact hypersurface immersed in a Euclidean space with constant mean curvature, which is stable.

In this paper we first prove that:

"The Catenoid and the Enneper's surface are the only orientable complete minimal surfaces in R 3 with index equal to one".

This gives a new characterization of these famous examples. Our second result gives a complete solution to the global stability problem in

the constant mean curvature case. Concretly we will prove the following fact:

"The only orientable complete surfaces with constant mean curvature in R 3 which are stable, are the plane and the sphere".

We also observe that a complete surface with constant mean curvature has

34

Complete minimal surfaces 35

finite index if and only if it is compact or a minimal surface with finite total curvature (examples of compact surfaces other than spheres are constructed by Wente [12]).

Finally we prove two properties of the Gauss map for complete surfaces in R 3 with non-zero constant mean curvature, which generalize a result Of Hoffman, Osserman and Schoen [9].

The methods that we use in this paper are based, for the most part, on Fischer-Colbrie's work [3].

Preliminaries

Let (M, ds 2) be a complete Riemannian surface. We denote by K, A and dA the Gaussian curvature, the Laplacian and the canonical measure corresponding to the metric ds 2. Given a smooth function q:M---~R we consider on M the operator L = A + q. We denote by Q the quadratic form associated to L, that is

Q(u, u) = - fM ( Lu)u dA = fM [l Vul2 - qu2] dA,

where u has compact support on M. For any bounded domain s in M, we denote by Ind (L, s the index of Q restricted to H~.2(I2). By definition, the index of L, Ind (L), is the supremum of the numbers Ind (L, s Given a minimal isometric immersion of M in R 3, dp:M----->R 3, we consider on M the Jacobi operator L = A + Io] 2, o being the second fundamental form of the immersion. We define the index of M as Ind (M) = Ind (L). If Ind (M) = 0 we say that M is stable.

If cp:M--~R 3 is a isometric immersion with constant mean curvature H, we consider also the Jacobi operator on M, L = A + Iol 2. Given a bounded domain s in M we define the signed volume enclosed by g2 as

(q,, N) dA

where N:M-+S2(1) is the Gauss map of M. As surfaces with constant mean curvature have critical area for variations

which preserve the signed volume, the operator L and the quadratic form Q, o have geometric meaning only for functions u ~ H, 2(g2) satisfying .[Mu dA = 0, see

details in [1]. So we define a new index for s Ind0 (L, I2), as the index of Q restricted to the space {u e HC~.z(I2)/.[nu dA = 0}. We define the index of M, as

36 FRANCISCO J. LOPEZ AND ANTONIO ROS

surface with H = const., by

Indo (M) = sup {Indo (L, s'2)/g2 bounded domain in M}.

Finally we say that M is stable, as surface with H = const., if Indo ( M ) = 0. Observe that for minimal surfaces we have defined two different indices, Ind (M) and Ind0 (M).

1. Minimal surfaces with index 1

Let M be an orientable complete minimal surface in R 3, and N:M---, sa(1) the Gauss map. If Ind (M) is finite then, from the result of Fischer-Colbrie [3], Theor. 2, M has finite total curvature, and so there exist a compact Riemann surface X, such that M is conformally equivalent to X - {p~, P2, �9 �9 �9 P~}, Pk e 2", k = 1 . . . . . n. Moreover the Gauss map extends to {Pl . . . . . p~} as a meromor- phic map N1:2"---~ sE(1), see Ossermann [10], Chapter 9.

Let ds 2 be a metric on 2" compatible with the Riemann structure. We define the operator L~ on 2" by

LI= AI + IVIN~[ 2, (1.1)

where A1 and VI are the Laplacian and the gradient of the metric ds~. From [3], Corol. 2 we have

Ind (LI) = Ind (M) (1.2)

we denote by Q~ the quadratic form associated to L~, i.e.

Q l ( u , u ) = - f U L l U d A l , for all UeHl.2(X)

Finally we will need the following slight extension of a result of Hersch [7].

LEMMA 1. Let p : X--~ R be a positive smooth function and t / / :X---~S2(1) a non-constant meromorphic map. Then there exists a conformal transformation g:SZ(1) ---, $2(1)) such that

f zp (g o ~ ) dAl = O,

where dA~ is the canonical measure associated to ds~.

Complete minimal surfaces 37

T H E O R E M 2. Let M be a complete orientable minimal surface in R 3. Then Ind (M) = 1 if and only if M is either the Catenoid or the Enneper's surface.

Proof. If M is the Catenoid or the Enneper's surface it is proved in [3], p. 132 that Ind (M) = 1.

Suppose now Ind (M) = 1. Then by (1.2) we have Ind (L1) = 1. Let p be the first eigenfunction of L1. Recall that P is positive everywhere. By hypothesis for all u e CI(,~) such that [xpu dA~ = O, we have

Q,(u, u) >- 0 (1.3)

the equality holding if and only if L~u = O. Let qt:2"._.~ $2(1) be a non-constant meromorphic map. Using Lemma 1 we

have a conformal transformation, g, of $2(1) such that ~ = g o gt'2"---~S2(1) satisfies

f pCItdA1 = 0 (1.4)

On the other hand

Q~(~, ~ ) = fzc(] v~qsl2-IV~N,12I~I2)dA~

= f~ (I Vl tPl2 - I V, Nll 2) dA,

= 8H[degree ( ~ ) - degree (N1)],

which combined with (1.3) and (1.4) say that

degree (t/t) = degree ( ~ ) => degree (N1), (1.5)

the equality holding if and only if A1~- t - IVxN1[2~=0. But as ~ is a meromorphic map we have A1 ~ + [ V~ ~[2 ~ = 0. So the equality (1.5) implies IV11~tl2 = I I71N1[ 2 everywhere.

Now we discuss the following cases: a) 2" is a sphere. Taking gtas the identity map, from (1.5) we conclude that degree (N1) = 1. So

taking into account Osserman [10], Chapter 9, we conclude that M is the Catenoid or the Enneper's surface.

b) genus (2') >- 1.

38 F R A N C I S C O .I. L O P E Z A N D A N T O N I O ROS

Recall that degree (N1) --> genus (2")+ 1, see Hoffman and Osserman [8], p. 75. If the inequality is strict, we construct via the Riemann-Roch Theorem a meromorphic map tp, with d e g r e e ( ~ ) = g e n u s ( 2 " ) + l , and (1.5) gives a contradiction.

If the equality holds, let p e2" be a regular point of NI which is not a Weierstrass point of _r. Again from the Riemann-Roch Theorem we obtain a meromorphic map, ~, of the same degree than NI which has p as a point of ramification of order genus (2") + 1. Then w ~ h e equality in (1.5) and so I~'1q~I2--IVINll 2 everywhere. But IVlqJ l2(p)=0 and IVINIIZ (p) 4:0. This is a contradiction and the theorem is proved.

2. Stable constant mean curvature surfaces

Let (M, ds a) be a complete Riemannian surface. Fischer-Colbrie has proved the following results.

T H E O R E M 3 [3]. I f the operator L = A + q has finite index on M, then there

exist a compact set C in M and a positive funct ion u on M with Lu = 0 on M - C.

T H E O R E M 4 [3]. Let C be a compact set in M and c a positive constant. I f

there exists a posit ive funct ion u on M such that Au - Ku + cu <- 0 on M - C, then

M is compact.

Theorem 4 is not explicitly stated in [3]. A consequence of this result is mentioned at the introduction of her paper. The proof follows from a simplifica- tion of the arguments given in Theorem 1 of [3]. For completeness we give some hints how the proof of this Theorem should be modified in order to yield Theorem 4.

Suppose that M is not compact. Following the proof of Theorem l in [3], and using the differential inequality A u - Ku + cu <= O, instead of inequality A u -

Ku <- 0 used in [3], we transform the least integral inequality of p. 127 in [3], into

f,7 [(~0 3 2 + 2WW" + c v / l ( u o r ) (s) as <- O, (2.1)

for any smooth function ~p : [0, o0)---, R vanishing for large s, with lp(0) = 0 where y(s) : [0, oo)---~ M - C is a certain curve paramel~rized by arc length.

Complete minimal surfaces

Given a > 0 we consider the function ~Pa : [0, ~) ~ R defined by

39

/-/s ~ p a ( s ) = s i n - - if O<=s<=a,

a

~Oa(s) = 0 if a<=s.

It is clear that (2.1) remains true if we take ~p = ~p~. Then we have

a [ H2 c0s2/-/s (c 2U2~sin2-~luo]/ds~O. f~L a-7 --a + -- a2 ]

As u 0 7 is posit ive, taking a large enough we obtain a contradict ion. So M is compact .

The following is the main result of this section.

T H E O R E M 5. Let M be a orientable complete surface with constant mean

curvature in R 3. Then M is stable (as surface with constant mean curvature) i f and

only i f M is a plane or a sphere.

Proof. It is clear that the p lane and the sphere are stable. Let L = z~ + Iol 2. Then we have easily that

1 + Ind0 (M) - Ind (L) (2.2)

So f rom the hypothesis L has finite index on M, then f rom T h e o r e m 3 there exists a posit ive solution of the equat ion Lu = 0 outside of a certain compac t set C in M. The Gauss equat ion says that �89 Iol z = 2 H 2 - K, so

0 = Lu = Au - Ku + (2H 2 + �89 1012)U ~-- A U -- g u '1- 2HZu,

on M - C. If the m e a n curvature is non-zero it follows by T h e o r e m 4 that M is compact . In this case f rom the result of Ba rbosa and do C a r m o [1] we conclude that M is a sphere .

If M is minimal by (2.2) we have the following al ternative: i) Ind (L) = 0, and so M is a plane, [4] or [2].

ii) Ind ( L ) = 1, and f rom T h e o r e m 2, M is the Catenoid or the E n n e p e r ' s surface.

We conclude the p roof of T h e o r e m 5 taking into account the following lemmas.

4 0 FRANCISCO J, LOPEZ AND ANTONIO ROS

L E M M A 6. The Catenoid is curvature).

Proof. The Catenoid is given by

unstable (as surface with constant mean

X(u , v) = (cosh v cos u, cosh v sin u, v), v ~ R, u e [0, 2 / / ]

So the induced metric and the Gaussian curvature are given by

ds 2 = cosh 2 v(du 2 + dv2), K = - 1 / c o s h 4 v

If the Catenoid were stable, then for any Lipschitz function f (u , v) = f ( v ) with compact support on R, satisfying

f ~ f ( v ) c o s h 2 v d v = O, ( 2 . 3 )

we will have

F F f ' ( v ) z dv >= (2/cosh 2 v) f (v ) z dv (2.4) o r a c

So the Lemma is proved if we can construct a function f, which satisfies (2.3) but not (2.4). Given 0 < a < b we consider the function f~,b:R ~ R defined by

f~ .b(v )=l if IV[=<a,

L , b ( o ) = (Iol - b)/(a - b)

fa.b(O) = 0 if b =< Ivl

if a<-ivl<-b

Observe that If',b(V)[ <-- 1/(b - a). For k =>0 we take f = 2f~.b --f,+k,b+k. If k = 0 or k = oo, the integral in (2.3) is positive or -oo respectively. So there exists k such that f verifies (2.3). We call this function fQ,b"

Then we have easily that f , , b ( v ) = l if Ivl<-a, I f 'b l<=2/(b--a) , and the subset of R on which f ' .b ~ 0 has length not greater than 4(b - a).

Putting ~.~ in (2.4) and taking limit when b goes to + ~ we obtain a contra- diction. Q .E .D.

L E M M A 7. The Enneper's surface is unstable (as surface with constant mean curvature).

Proof. Via the Weierstrass representation, the Enneper ' s surface is para- metrized by the whole complex plane.

Complete minimal surfaces

The induced metric and the Gauss curvature are given by

41

(As 2 : [�89 + Izl2] z Idzl 2, K = - [4 / (1 + IzlZ)2] 2

In a similar way than in Lemma 6, we consider functions in polar coordinates f(r, O)=f(r ) . If the Enneper 's surface were stable, then

fo ~ fo ~ 8f(r)2r f '(r)2r dr >= (1 + r2) z dr

for any LipschRz function, vanishing for large r, such that

f0 =f(r)(1 + r2)2r dr = 0

Using an argument as in Lemma 6, we get a contradiction. Q.E.D.

Now we will prove some simple but relevant results.

PROPOSITION 8. Let M be a complete orientable surface immersed with constant mean curvature in R 3. Then Indo (M) is finite if and only if M is either compact or a minimal surface with finite total curvature.

Proof. If Indo(M) is finite, reasoning as in the proof of Theorem 5, we conclude that M is compact or minimal. In the minimal case we have from [3] Theor. 2, that M has finite total curvature.

The nontrivial part of the converse assertion follows also from [3], Theor. 2, and the inequality Indo (M) _-< Ind (M), which holds clearly for minimal surfaces. Q.E.D.

The following result is an interesting consequence of Theorem 4.

C O R O L L A R Y 9. Let M be a complete, non compact, Riemannian surface, and q : M - - , R a smooth function on M such that q>-c everywhere, for some positive constant c.

Let u be a solution of the equation Au - Ku + qu = 0 on M. Then u-I(0) is a non-compact set in M.

Proof. On M - u-I(0) , lul is positive and satisfies A lul - K lul + c lul -<0. If u-l(0) is compact, modifying lul in a bounded neighborhood of u-I(0) we get a

42 F R A N C I S C O J. L O P E Z A N D A N T O N I O ROS

positive function on M which satisfies the above differential inequality outside of a compact set. So theorem 4 gives a contradiction. Q.E .D.

The study of the Gauss map is one of the fundamental problems in the classical theory of surfaces. For surfaces with constant mean curvature we have the following result.

T H E O R E M 10. Let M be a complete, non-compact, surface immersed in R 3 with non-zero constant mean curvature. Let N be the Gauss map of M, N:M----~S2(1). Then N - I ( S 1) is a non-compact set in M for any equator S l in S2(1).

Proof. We recall that A N + [o[2 N = 0 on M, or equivalently, for any vector a in $2(1) the function u = ( N , a ) satisfies the equation A u - K u + ( 2 H 2 + �89 Iol2)u = 0. If S ~ is the equator in S2(1) orthogonal to a, then N - I ( s 1) = u-l(O).

So the result follows from Corollary 9. Q .E .D.

Note that if X is a Killing vector field in R 3, the function u = (N, X ) on surfaces with constant mean curvature satisfies the equation Au + [o]2u = 0. So for these functions we have the corresponding version of Theorem 10.

Finally, we state the following extension of a result in [9].

T H E O R E M 11. Let M be a complete, non-compact, surface immersed with non-zero constant mean curvature in R 3, and let N : M ~ $2(1) be its Gauss map. I f N ( M - C) lies in a closed hemisphere of $2(1), for some compact set C in M,

then M is a circular cylinder.

Proof. Suppose that the hemisphere is given by {x �9 S2(1)/(x, a) >= 0}, with a in Sz(1), Then the function u = (N, a ) is non negative and satisfies the equation Au +lolZu = 0 on M - C . From the maximum principle the function u is constantly zero or everywhere positive on M - C. In the first case we conclude using the unique continuation property that u = 0 on M, and so M is a circular cylinder, see [9].

In the second case, Theorem 4 gives a contradiction. Q.E .D.

Added in August o f 1987. A. M. da Silveira and B. Palmer have obtained independent proofs of Theorem 5. Also S. Y. Cheng and J. Tysk has announced a proof of Theorem 2 in the embedded case.

REFERENCES

[1] L. BARBOSA and M. DO CARMO, Stability of Hyper:~urfaces with Constant Mean Curvature. Math. Z. 185, 339-353, 1984.

Complete minimal surfaces 43

[2] M. DO CARMO and C. K. PENG, Stable complete minimal surfaces are planes. Bull. Amer. Math. Soc. 1, 903-906, 1979.

[3] D. FISCrtER-COLBRIE, On complete minimal surfaces with finite Morse index in three manifolds. Invent. Math. 82, 121-132, 1985.

[4] D. FISCHER-COLBRIE and R. SCHOEN, The structure of complete stable minimal surfaces in 3-manifolds o f non-negative scalar curvature. Comment. of Pure and Appl. Math. 33, 199-211, 1980.

[5] R, GULLIVER, Index and total curvature of complete minimal surfaces. Geometric measure theory and the calculus of variation Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 207-211 1986.

[6] R. GULLIVER and H. B. LAWSON, The structure of stable minimal hypersurfaces near a singularity. Geometric measure theory and calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Syrup. Pure Math. 44, 213-237 1986.

[7] J. HERSCH, Quatre propidtds isoperimetriques de membranes spheriques homog~nes. C. R. Acad. Sci. Paris. 270, 1645-1648, 1970.

[8] D. A. HOFFMAN, R. OSSERMAN, The Geometry o f the generalized Gauss map. Memoirs of the American Mathematical Society, 236, 28, 1980.

[9] O. A. HOFFMAN, R. OSSERMAN and R. SCHOEN, On the Gauss map of Complete Surfaces of Constant Mean Curvature in R 3 and R 4, Comment. Helvetici, 57, 519-531 1982.

[10] R. OSSERMAN, A survey of minimal surfaces. New York Van Nostrand Reinhold 1969. [11] A. V. POGORELOV, On the stability of minimal surfaces. Soviet Math. Dokl. 24, 274-276 1981. [12] H. C. WENTE, A counterexample to a conjecture by H. Hopf. Pacific Journal of Math., 121,

193-234 1986.

Departamento de Geometria y Topologia Facultad de Ciencias Universidad de Granada 18071 Granada (Spain)

Received April 2, 1987