Ann. Global Anal. Geom.Vol. 8, No. 3 (1990), 217-226

Spacelike Constant Mean Curvature Surfacesin Pseudo-Riemannian Space Forms

BENNETT PALMER

0. Introduction

This paper is concerned with the study of spacelike constant mean curvature surfacesin three-dimensional pseudo-Riemannian manifolds of constant sectional curvature c.Below we define models for these manifolds which we denote by 23(c). Although theintrinsic geometry of a surface M c 3(c) differs radically from that of a surface in aRiemannian space form, the structural equations in both cases are formally very similar.For surfaces in flat Minkowski space 3 (0) this was noted in [M]. This formal similarityis used in Section 2 to extend a theorem due to H. B. Lawson via which surfaces ofconstant mean curvature in one space form isometrically correspond to those in another.

The main result of this paper is the following theorem.

Theorem I. Let R be a compact Riemann surface of genus g and let Q be a holomorphicquadratic differential on R. Let h and c be real numbers. Let R be the simply connecteduniversal cover of R. If

(i) g > 1, andh2 -c > 0 ,

(ii) g = 1, Q O and h2 - c > 0,

(iii) g = 1, Q 0 andh2 - c = 0,

or

(iv) g = Oandh2 - c < 0,

then there corresponds to Q a unique (modulo isometries of 3 (c)) one parameter familyof spacelike conformal immersions

R -, 3(C).

The surfaces thus obtained have constant mean curvature and are complete in the inducedmetric. The surfaces in each family are pairwise isometric.

In [R] J. Ramanathan has shown that a complete, spacelike surface in 3 (1) withh - 1 = const < 0, is a totally umbilic 2-sphere.

The proof of Theorem I involves the solution of a non-linear P.D.E. on R which isthe Gauss equation of the surface (Theorem III). In general the surfaces obtained viaTheorem I are not compact (Theorem IV). Complete surfaces with constant meancurvature in 3 (0) have been constructed by A. Treibergs, [T], by another method.

15 Annals Bd. 8, Heft 3 (1990)

218 B. Palmer

The main results of this paper were obtained while the author enjoyed the hospitalityof the Sonderforschungsbereich 256 of the University of Bonn.

1. Models for Pseudo-Riemannian Space Forms

On R consider the symmetric bilinear form

I--s

<x, y)> = E xiyi - E XiYi,1 n-s+l

x = (x1, ..., xI), Y = (Yl, " , Yn)

The pair (Rn, (,)>) will be denoted by R:. Define V2 (c) as follows:

(i) c = 0, Z3(c) = R, that is, three-dimensional Minkowski space.

(ii) c > 0, E3(c) = {x E R I <x, x> = r2}, where c = 1/r2 .

(iii) c < 0, £3(c) = {x E R4 I (x, x> = -r 2 }, where r2 = - 1/c.

In each case we consider V3(c) only with the induced pseudo-metric from R:. It is shownin [W] that Z3(c) is a complete 3-dimensional, pseudo-Riemannian manifold of constantsectional curvature c and signature (+, +, -).

The Z3(c)'s will be considered as the pseudo-Riemannian analogues of the standardRiemannian space forms: R3, the three sphere and the three-dimensional hyperbolicspace. Although they are not in general simply connected, they are defined as quadraticvarieties in a vector space.

2. Surface Theory in 3(c)

Call a plane it c R spacelike if the restriction of (,)> to 7l is positive definite. A smoothlyimmersed surface M2 c R will be called spacelike if the tangent space at each point isspacelike.

Let

X: M - (c) c R (2.1)

be a spacelike orientable surface. Introduce an isothermal coordinate z = u + iu2 onM taking values in a simply connected Riemann surface R. The metric induced by Xhas an expression

ds2 = e Idzl2 (2.2)

for a smooth function = (z) on R.Since M is orientable, we may continuously define a normal vector field it along M

which satisfies

(2.3)(<t, )> = - 1

Spacelike Constant Mean Curvature Surfaces in Pseudo-Riemannian Space Forms

As in the Riemannian case, the differential dl defines a self-adjoint endomorphism ofthe tangent space at each point. Define the second fundamental form at p E M:

np(x, y) = - dr(x), y>; x, y TpM. (2.4)

In terms of the chosen coordinate we can write

n = Re {(p dz ® dz + h e dz d} (2.5)

where

h = 2 trace (di&) (2.6)

is the mean curvature of the immersion. The quantity p dz dz defines an invariantquadratic differential on M.

The quantities defined above appear as coefficients in the structural equations for theimmersion. Letting subscripts denote partial derivatives, we have on R

1Xz = zXz--2 'p XZZ = -XI - 2 2

2

(*) 1 el'g, = -elhrl --

2 2r2

r/1 = -h - p e- ° Z

where = 0, + 1 or - 1 if respectively c = 0, c > 0 or c < 0.By taking third derivatives using (*) and equating the mixed partials, one obtains the

important integrability equations of Gauss

ea l- e-(G) QZ = h2_-Ip12 - -c-

2 2 2

and Codazzi

(C) pf = e° hz .

The intrinsic curvature x of the metric ds2 is given by

x = -2 e zZ (2.7)

so that (G) can be written

lJp2 e- 2 = (h2- c) + X. (2.8)

When h = const, equation (C) reduces to

¢ = 0, (2.9)

i.e., p is a holomorphic quadratic differential on R. The exploitation of this fact isessential in the study of constant mean curvature surfaces in both Riemannian andpseudo-Riemannian space forms.

15'

219

220 B. Palmer

When (2.9) holds, equation (G) takes the form

e = F(z, Q)

with F real analytic in both its arguments. It follows that a smooth solution of (G) isreal analytic. If such a solution exists on R, then all coefficients of the system (*) arereal analytic and there exists locally a real analytic solution X of the system (*). Usingthe simple connectivity of R and an easy argument involving analytic continuation onecan produce a global solution

X: R - 3(c).

As in the case of surfaces in Riemannian space forms it can be shown that the solutionis unique modulo isometrics of Z3(c). The key fact that is needed is the transitivity ofthe isometry group of RS on orthonormal frames at distinct points (see [0], p. 113).

As a consequence we have

Proposition 2.1. Let (p be holomorphic and assume is a global smooth solution of (G)on R. Then there exists a unique (modulo isometries of 3 (c)) solution of (*) on R.

Using Proposition 2.1 we easily obtain:

Theorem II. Let

X: Z -*3(c)

be a conformal spacelike immersion with constant mean curvature h. Let h, c be realnumbers with

h2 - c = h -c. (2.10)

Then there exist a 2i-periodic family of local isometric immersions

Xo,c1: (R, ds2) - Z 3(c1 ), 0 e [0, 2t] (2.11)

with constant mean curvature h,. This family contains all extensions of smooth isometricimmersions into Z3 (C,) with constant mean curvature h,.

Proof. If X is given as a solution of (*) with data e, h, p, then note that (G) holds if h,c are replaced by any pair h, c such that (2.10) holds. Also the Codazzi equation(p = 0 continues to hold if is replaced by ei° p, 0 e R. The final statement follows byProposition 2.1.

Remark. In [P] the author gives a Lie Theoretic interpretation of the special case c = - 1,c = 0.

3. Construction of Complete Surfaces

Let R be a compact Riemann surface of genus g, i.e., a 2-dimensional manifold with afixed conformal structure. It is well known that R may be endowed with a metric dsOof constant curvature xO which is compatible with the given conformal structure. Assumethis has been done and write R = (R, dso).

Spacelike Constant Mean Curvature Surfaces in Pseudo-Riemannian Space Forms

Introduce a global complex coordinate z on R. Then the metric defined above hasan expression

dso = e dzl2 . (3.1)

If Q denotes any holomorphic quadratic differential on R, then Q has an expression

Q = p dz dz (3.2)

with

(P = 0. (3.3)

Each such Q has an invariant norm defined by

lIQl12 = oP2 e - 2' . (3.4)

We investigate under what conditions there exists a conformal metric

ds2 = e ds2 (3.5)

on R, satisfying the Gauss equation (G) for a given Q = qp dz dz.

Proposition 3.1. Let Q = op dz dz be as above. Let h and c be real constants and u a smoothsolution of

Au + 211Q112 e - 2(h 2 - c) eu = 2xO. (3.6)

(3 _ 4 e -e a: a, is the Laplace-Beltrami operator on R.) Then the metric ds2 defined by(3.5) satisfies (G).

Proof. Using (2.7) with = u + a, one sees that ds2 has intrinsic curvature

e-u e-ux = - (u + a) = - Au + e -u x .

2 2

Using this together with (3.4), (3.6) can be rewritten as

oI¢2 e- 2(u+a) = h2 - C + x

which is the same as (G).

Remark. In another setting K. Uhlenbeck [U] studied the Gauss equation of an immersedsurface by writing the metric as a multiple of ds0.

Theorem III. Let R be a compact Riemann surface of genus g. Let Q be a holomorphicquadratic differential on R. Let h and c be real numbers. There exists a unique smoothsolution of (3.6) on R in any of the cases (i) through (iv) of Theorem I.

Remark. The proof of Theorem III employs a technique used by Melvyn Berger [B] toproduce a metric of constant curvature xO = -1 on R with g > 1. We will need thefollowing result.

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222 B. Palmer

Theorem. (Hadamard). Let X be a Banach space and F: X - X a Frchet differentiablelocal homeomorphism with derivative DF. Suppose there exists fl E R such that

II(DF(x))- I < f

holds for all x E X. Then F is a global homeomorphism.

A proof can be found in [B].

For the remainder of this section, unless stated otherwise, all computations will bedone using only the metric ds' on R. W, 2 will denote the space of real valued functionsu on R such that u and its gradient Vu are square integrable. W, 2 is a Hilbert spacewith inner product

<(, v> = (VU Vv + uv) (3.7)

and associated norm 11,(1,2). Let

E = {u C W1,2 I u = O).For u e E

i-x l IVul2 > i u2 (3.8)

where is the first non-zero eigenvalue of the Laplacian. Therefore

<U, V>E = Vu Vv

defines the same topology as (3.7).A result which can be found in [A], p. 63, states that there exist constants c and ,

such that for w e W,2

few •_c cexp [,u II wl(1, 2)] (3.9)holds.

Proof of Theorem III. Introduce implicitly defined operators L and N by

(Lu, v> = VuVv

<Nu, v> = S (2A eu - 2 QIl2 e-") v

where A = h2 - c.

Lemma 3.1.(i) L is a continuous self adjoint map of W, 2 to itself.

(ii) N is a C' (continuously Frechet differentiable) map of W, 2 to itself.

Proof of Lemma 3.1. Part (i) is clear. Part (ii) follows from the following statement: Forip E C (R) and a E R the map F defined by

<Fw, v> = q eW . v

is a C' map of WI, 2 to itself.

Spacelike Constant Mean Curvature Surfaces in Pseudo-Riemannian Space Forms

Formally compute the derivative DF to get

(DF(w) v1, v2> = f 1Wp ew v1v2 , vlv2 E W1,2

Since Wp is bounded above, it is enough to give a uniform bound on

S eMw V2

for v with IIvll(1, 2) _ 1. Estimate

e v2 = eaw + I exp (w + 2 log v)R (v

2< 1) {v

21)

- I e ' + j exp (w + 2v){v

2< 1 } (

22 1)

< e + (e) 1 /2 ( 4 1/2

Applying (3.9) to each of the integrals above gives the required bound.

The inequality (3.9) can also be used to show that

w DF(w)

is a continuous map of W, 2 into the space of bounded linear functionals.

Returning to the proof of the Theorem, we decompose W1,2 as follows.

W1,2 = ker L E. (3.10)

Let P denote orthogonal projection of W1, 2 onto E. Corresponding to the two summandsin (3.10) the equation (3.6) can be written in two parts. For u = w + m a solution of(3.6) with w = Pu, m E ker L, we have

Lw + PN(m + w) = Pf (3.11)

where f is implicitly defined by

<f, v>= -2 xv .

Also

e-m 211QI 2 e- w - em 2A ew = 2 xo = 4s (3.12)

must hold where = 2 - 2g. Conversely a solution of (3.11) and (3.12) solves (3.6)modulo regularity.

Using (3.12), m can be determined uniquely from w. Write (3.12) as

r/x - qx = 4£

where x = e. In cases (i) and (ii), r, q and hence w determine a unique x as the positiveroot of a quadratic with discriminant (47) 2 e2 + 4rq. Cases (iii) and (iv) will be handledseparately. Use x to define m = m(w). It is now enough to produce a solution of (3.11).

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224 B. Palmer

To this end we apply Hadamard's Theorem with E = X and

F(w) = Lw + PN(m(w) + w).

For v e E, one has

<DF(w) v, v> = <DL (w) v, v> + DPN(m(w) + w) v, v> .

Note that

<DL (w) v, v> = d [S V(w + tv) Vv],=o = Vv Vv .

Also

d<DPN(m(w) + w) v, v> = - (<PN(m, + w,), v>)=o

d<DPN(m(w) + w) v, v> = d (<N(m, + w,), v>),=o

= d-(S (2A em ewt- 2 IIQ2I e-"t e-w') v)t=,.dt

Letting "dot" denote - ()t=O, the right hand side of the above equations is

S (2A ew+m + 2 Q12 e- (W+))v + (2A ew+m + 2 Q112 e-(W+m)) v2. (3.13)

In order to estimate this expression, implicitly differentiate (3.12) with respect to t to obtain

-t ( (2A ew+m + 2 IIQII2 e -(w+m)) = (2A e + "m + 2 llQ112 e - (w+m) ) .

Using this we solve for it, substitute into (3.13) and collect terms to obtain

<DF(w) v, v> = fVv Vv

- (5 (2A ew+m + 2 IIQi12 e- (w+ ')) v)2

x ( 2A ew+m + 2 IIQI12 e- (w+") - 1

+ (2A ew+m + 2 IIQII 2 e -(+m)) v)2 .

The sum of the last two terms is non-negative. This may be seen by cross multiplyingand applying H6lder's inequality. It follows that

<DF(w) v, v> _ Vv Vv = IIvI12.

The Lax-Milgram lemma implies DF(w): E -* E is invertible and

II (DF(w))-'II < 1

Spacelike Constant Mean Curvature Surfaces in Pseudo-Riemannian Space Forms

so that Hadamard's Theorem can be applied to find a unique w such that u = w + m(w)is a weak solution of (3.6).

Rewrite (3.6) in the form

Ju = .

The estimate (3.9) may be used to show p e L P for all p > 1. By using properties of theGreen's function of R ([A], pg. 108), one may obtain a classical solution ia equal to ualmost everywhere on R.

In case (iii), xo = 0, so u = 0 is the unique solution. In case (iv) it is well known thatQ 0 and xo > 0. There is a unique solution which is the constant ot given by

-2(h 2 - c) e = 2x0.

Remark. Immersions with Q 0 are totally umbilic. In this case the form II is a(function) multiple of the identity and the curvature x is constant.

Remark. It is well known that the dimension of the space of holomorphic quadraticdifferentials on R is 0 if g = 0, 1 if g = 1, and 3g - 3 otherwise.

Proof of Theorem I. Using Theorem II we obtain a smooth solution of (3.6) on R. Welift the metric ds2 = e dso to the simply connected universal cover / and denote thelift again by ds2. Since u is bounded below (, ds2 ) is complete. Proposition 1.1 may beapplied to produce the desired immersions.

In general the surfaces obtained from Th. I are not compact. Note that when c < 0Theorem I implies the existence of complete immersions in 3(c) with h = 0.

Theorem IV. Let c < O. Then there is no compact, spacelike orientable surface with h 0in 2 3(C).

Proof. Without loss of generality, by rescaling if necessary, c = - 1. Suppose

R - M c 3 (-1) _ {X E R Ix + x2-X2 _2 = -1}

is an immersion with h = 0 and M compact. From (*) we obtain (using the Laplacianon M)

AXj = 2Xj, j 1 ... , 4,so that

/Xj = 2 (xj AXj + IIVxjll2) > 4.

Clearly

X3 + X4 _ 1

holds on M. Therefore

A(XI + X4) 4

and integration over M gives a contradiction.

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226 B. Palmer

4. Closing Remarks

Let u be a solution of (3.6) with h2 - c > 0. The linearization of (3.6) at u is the selfadjoint, elliptic operator

Lyp = Aip - 2(11Q112 e- u + (h2 - c) e)wp.

The first non-trivial eigenvalue of the problem

Lyp + AW = 0

is characterized by

A = inf (IVpl 2 + 2Q(lIQ 2 e -u + (h2 - c) e) 2)weC°(R) p2

It is clear that solutions of (3.6) are stable in the sense that A > 0. For this reason onewould expect that a variational proof of Theorem III is possible.

Surfaces with h = const in 3(c) may be characterized variationally as critica for thearea functional with a volume constraint. The stability of solutions of (3.6) is closelyrelated to the fact that the surface is a local maximum for this variational problem. Thisis in sharp contrast to the case of surfaces in Riemannian space forms. For example, ifM2 c S3 is a compact minimal surface, any solution of the analogous Gauss equationis necessarily a saddle point.

References

[A] AUBIN, T.: Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag,New York 1982.

[B] BERGER, M. S.: Nonlinearity and Functional Analysis. Academic Press, New York, 1977.[M] MILNOR, T. K.: Harmonic Maps and Classical Surface Theory in Minkowski 3-Space.

Transactions of the A.M.S. 280, 1 (1983).[O] O'NEILL, B.: Semi-Riemannian Geometry, Academic Press, New York, 1982.[P] PALMER, B.: On a Theorem of Lawson (in preparation).[R] RAMANATHAN, J.: Complete Spacelike Hypersurfaces of Constant Mean Curvature in de Sitter

Space, Indiana University Math. J. 36, No. 2 (1987).[T] TREIBERGS, A. E.: Entire Spacelike Hypersurfaces of Constant Mean Curvature in Minkowski

Space, Invent. Math. 66 (1982), 39-52.[U] UHLENBECK, K. K.: Closed Minimal Surfaces in Hyperbolic 3-Manifolds, in Seminar on

Minimal Submanifolds, Princeton Univ., 1983.

BENNETT PALMER

University of CaliforniaSanta Cruz, CA 95064USA

(Received June 27, 1989)