surfaces in lorentzian hyperbolic space

Ann. Global Anal. Geom.Vol. 9, No. 2 (1991). 117-128

Surfaces in Lorentzian Hyperbolic Space

BENNETT PALMER

0. Introduction

In this paper we study spacelike surfaces in the 3-dimensional hyperbolic, Lorentzianspace form H3. For an immersion of a surface

X: M H (0.1)

we define a Gauss map

:M - F, (0.2)

where F is an appropriately defined Grassmann manifold. This Grassmannian, with itsnatural pseudo-Riemannian structure, admits a product structure

F Dx D, (0.3)

where D is essentially the 2-dimensional hyperbolic space of constant curvature -2.This development closely parallels the decomposition of the Grassmannian of 2 planesin E4 as a product of 2-spheres, a fact which motivates several results concerning 7. Forexample, if relative to (0.3) is given by y = (7+, ' ), then we show that 7+ and -_have equal energy densities.

In [P] the author constructed a large number of examples of complete spacelike surfacesof constant mean curvature (h-surfaces) in H 3. Specializing to h-surfaces, we show that themap ;' is harmonic, a generalization of the Ruh-Vilms Theorem. Consequently y ± are har-monic. Motivated by results of Hoffman, Osserman and Schoen [HOS] we study the geom-etry of M under the hypothesis that A + (ori _) omits a suitably large subset of D. In partic-ular we give an explicit upper bound, depending only on and h, on the radius of a geo-desic disc in M under the assumption that y + (M) lies in geodesic disc of radius z in D.

In the final section we construct a B/cklund transformation for h-surfaces in H. In[P] it is shown that the one parameter ,,associate family" of isometric h-surfaces can bederived from a given one, as in the case of h-surfaces in Riemannian space forms. TheBacklund transformation allows for the construction of these isometric surfaces involvingonly solution of first order linear systems of p.d.e.'s and quadratures.

1. A Pseudo-Riemannian Grassmannian

Let E denote the vector space R4 together with the product

<x, y> = x1y + x2Y2 - X3 3 - 4y4 (1.1)

118 B. Palmer

Let A2 = A/2(E) denote the second exterior algebra equipped with the product, againdenoted by (,) >, induced by (1.1). For generators u A v, U' A ' of A2 ,

<U A V, U' A V'> = U, U'> <V, V')> - U, V') U', V' . (1.2)

Define the Hodge star operator

*: A2-* A2 (1.3)

by the usual formula. If {ej} is the usual basis of R4 then

<A, B>el A ... A e = A A B; A, Be A2 . (1.4)

In terms of the standard basis {ei A ej}l<i<j<4 of A2, * is given explicitely by

el A e2 e3 A e4 , el A e3 e 2 A e4 , e A e4 -e 2 A e3 (1.5)

and *2 = id. The operator * induces a spectral decomposition

A2 = A2 A2 (1.6)

where

A2+ ker (* - (id.)). (1.7)

Choosing the bases of A+ given by

2 (e A e2 e3 A e4 ), (el e3), (el e4 e2 e3) (1.8)

one sees that A2_ is isometric to 3-dimensional Minkowski space with signature(+, , )

In /12 consider the variety F defined by the two conditions

(i) <A,A> = , (1.9)

(ii) <A, * A> = 0.

Writing A = a aije, A ej, one sees that these are equivalent toi<j

(i)' a22 - a23 - a24 - a 2 3 - a224 + a3 4 = 1, (1.10)

(ii)' al2a34 - al 3a2 4 + al 4a2 3 = 0.

The condition (ii)' is the well-known decomposability condition which implies A E F isexpressible as A = u A , u, v c E2. The first condition (i)' then implies that the restrictionof (1.1) to the plane spanned by u and v is either positive or negative definite. Denoteby F+ the subset of F consisting of those planes A with , >IA positive definite. It canbe shown that

r+ = {A E r I a22 > a 4}. (1.11)

For A e F+ denote the projections onto A2 by A,. Clearly A, = (A + *A) andtherefore by (1.9),

<A+, A+> = 2 (1.12)


Using the bases appearing in (1.8) we write

A+ = (Ul, u2 , u3 )

(a1 2 + a3 4, a13 + a2 4 , a1 4 - a23 ),

(1.13)A = (v1, V2 , 3 )

-2 (a1 2 - a3 4 , 1 3 - a2 4 , a 14 + a2 3)

1 1By (1.11) u1 > -and vr > --

1V2 1V2It follows that there is an isomorphism

F+ (D, -dP 2) x I). - dp 2), (1.14)

where (D, -dP 2 ) is a two dillcnsional hyperbolic space equipped with the negative ofthe Riemannian metric of constant curvature -2.

2. Surfaces in H3

In E4 consider the variety H3 defined by

H = {y E I (y, y> = -1}. (2.1)

It is shown in [W] that H3 is a Lorentzian manifold with constant sectional curvature-1. In fact H3 is essentially isometric to SI (2, R) equipped with the bi-invariant Killingform. Let R be a Riemann surface and let

X: R M c H 3 (2.2)

a conformal immersion of a spacelike, orientable surface M. If z is a complex coordinateon R, then the metric induced by can be expressed

ds2 = e Idzl2

Since M is orientable we can choose a timelike normal field r which is tangent to H3,satisfies <(r, ) = - 1, and -2i e- zQ A , A A gives the usual orientation of R4.The second fundamental form of M can be expressed by

- <dl, dZ> = 2 Re ( dZ2 + he dz d}

where h is the mean curvature function on M. The Gauss and Codazzi equations takethe form

l102 e 2 = h2 + 1 + k, (2.3)

119

(P = e hz . (2.4)

120 B. Palmer

We refer the reader to [P] for details. Combining (2.3) with a standard formula for thecurvature k of the metric ds2 yields

eo e-0Qez = (h 2 + 1)-- I1p12

2 2

One has the structural equations for the immersion X,

(i) X = zXz- 212

eQ eo

(ii) Xzz = - - h--q, (2.5)

(iii) = -hx - 'p e- Xz

For future references we remark that using the above one finds that iq satisfies

(pJ2 2 e\) eqz2 = - hz -he- + 2 e h X. (2.6)

2 2/ 2

3. The Gauss Map

For as above we define the Gauss map

y: M -- r+ (3.1)

by

Y = -2ie-z A XZ. (3.2)

It is easily checked that y is independent of the choice of coordinate z. Because of theresults of part 1 this mapping has many properties analogous to those of the Gaussmap of a surface in the Euclidean space E4. In that case the Grassmannian of 2-planesin E4 is a direct product of Euclidean 2-spheres. The interested reader should comparethe following result with those of Hoffman and Osserman [HO].

For a smooth map

f: M (N, do2 )

where (N, da2) is a pseudo-Riemannian manifold, we define

f* d 2 2Re (f* d 2)(2O')dz 2 + IJdfll2 e-dzd} (3.3)4

For y as above we denote by

7 : M - (D, -dP 2) (3.4)

the projection onto each factor of (1.14).


Proposition 3.1

(i) lld,112 = -2(2(h 2 + 1) + k),

(ii) IIdi'+112 = 1id - 112 I= d7H2,

(iii) '* (-dP2 ) = e y* (- dp2 ),

where e h + h-i

Proof: It follows from (2.5) that

(2 1 e )71 = 2ie-e ; A -- 2 heeZ A q + - z A ) (3.6)ee x,). (3.6)

(i) follows since

lid712 = 4 e- ( ?- (3.5)

Because *' =X A , one finds again by (2.5),

· 7r! = X, A t1 + h A + e-e ez A Xz (3.7)

so that <T(, * 7_- = 0. Therefore

<(a+, a8,±) = ¼ ('K + * _, ¢Z + *'z> = <(!, Tz> (3.8)

and (ii) follows.Using * z = (*r), (3.6) and (3.7) we find

:, * 'z> = -irp i z, Tz> = hpq (3.9)

so that

(.,* (-dtp2))(2.' = 0<aZY + aaz +

= 4 <, z ·*,z * i ,>

= (z' Yz> ±< * { ,»Z ,,>)

= ( p(h ± i).

Theorem 3.2. Let M c H3 be a spacelike immersion. If the mean curvature h is constant,then is a harmonic map. Consequently 7± are harmonic.

Prooqf: The theorem will follow by showing that satisfies

AI' = - d3 ll2 y + 2(*dh)# A (3.10)

where * is the Hodge operator of M and # is the usual duality map which raises indices.By (3.6) we have

'i' = 2(ip e-e r A Xz - i A + ihXz A q)

= ip ee * A X + ih2zX A + ((p e 2 + 1 + h2 )

(-iX- A Z)

121

122 B. Palmer

Using the Codazzi equation T. = e hz and the Gauss equation l2 e-2 e = h2 + 1 + k,(3.10) follows since = 4 e- 2 aa,.

When h = const., (3.10) reduces to the Euler-Lagrange equation for the variationalproblem,

f Idylld =0.

By considering variations of y tangent to one of the factors of (1.7), the harmonicity ofy+ follows from that of y.

Remark. When h = const., it can be shown that y, arises as the Gauss map of a local

isometric immersion of M into E with constant mean curvature Ih + 1.

Proposition 3.3. y is conformal iff M is maximal (h O) or M is totally umbilic. y ± areconformal iff M is totally umbilic.

Proof. From (3.6) we obtain

<yz, y> = -hp

so that y is conformal iff h 0 or p = 0. The latter is the totally umbilic case. Thesecond statement follows using (3.9).

The next result investigates the geometry of M under the condition that y, omits asuitably large subset of D.

For xo M let B,(xo) denote the geodesic ball of radius r centered at x. Similarlyfor Yo E D, let D,(yo) denote the geodesic ball of radius z.

Theorem 3.4. Let M c H3 be a spacelike h-surface. Assume that B,(xo) c M and that

y,7+(M) c D,(yo) (3.11)

for some Yo E D. Then

r2 a2 (h ( ) exp [2(ch (/) - 1)], a 2.81 . (3.12)

A similar statement holds for y_.

The proof of the Theorem 3.4 is based on the following result which is of independentinterest.

Theorem 3.5. Let M be a two dimensional Riemannian manifold. Let p e M such thatB,(p) c M. Assume the curvature k is bounded below by a negative constant k. Also assumethat there exists a smooth function u on Br(p) satisfying

Au > 1, (3.13)

0 u u < o u, = const. (3.14)

Surfaces in Lorentzian Hyperbolic Space 123

Let ;1 = 21 (A, Br(p)) denote the first Dirichlet eigenvalue of the Laplacianfbr Br(p). Then

a2

2. < e2 "k '-u ) , a 2.81, (3.16)r2

(u - u) - < < (3.17)

and consequently

r2 _ (u - u) e2~(u-a ) . (3.18)r

Proof: Let d 2 denote the conformal metric

d 2 = e2ku dS2

By a well-known formula, the curvature k of ds 2 is

k = -e-2ku (kAu - k) > O0 (3.19)

For convenience let B(r, p) = B,(p). Denote quantities on (B(r, p), dg 2) by a tilde. Weclaim

B(r ek", p) c B,(p). (3.20)

To see this let r' be curve in (B(r, p), d 2) of length . Then

[= d = e dx > e - ds e 1.

The inequality I > e I implies (3.20).Since k 0, we may apply an eigenvalue estimate of M. Gage [G] to obtain

a2 a2

2,(,, B( = re 2 k , p)) < < - e -2k. (3.21)

(r3 = r2

Using (3.20) and a well-known monotonicity property of eigenvalues, one obtains,

2k2I(,, B(r, p)) < (A, B(r e, p)) < e 2 . (3.22)

It remains to compare A = .1(J, B(r, p)) with ). J = ;il(, B(r, p)). Let f be a solutionof

Jf + 1 f = 0 in B(r, p),

f = 0 on B(r, p).

Since J = e-2,U J, f satisfies

Af + 1, e2ku f = 0 in B(r, p),

f = 0 on B(r, p).

124 B. Palmer

Consequently

0 = f IVfl2 -- e2ku f2 * 1B (r, p)

f lVfl2 - e2krf 2 *1B(r, p)

- (2,~ _ ,e2ka) f f2 *1.B(r, p)

It follows that

2(A, B(r, p)) < e21", l(J, B(r, p))

a2a< e2k(u-a)- r2

completing the proof of (3.16).To show (3.17), let tp > 0 be an eigenfunction belonging to A1. Let g(x, y) denote the

(positive) Green's function for B(r, p). Then

tp(x) = 21 f (y) g(x, y) * l(y)

and hence

(x) A,1 Il11ll g(x, y)* l(y).

Choosing x such that Il(x) = 1V1 one obtains

1 < Al max f g(x, y) * l(y) = Al max v(x). (3.23)X x

where v(x) solves

lv = -1 in B(r, p),

v = 0 on 8B(r, p).

Note that

A(u + v) > O0 in B(r, p)

and on B(r, p)

u+v = u < .

By the maximum principle applied to the subharmonic function u + v,

v < u-u < -u in B(r, p).

Returning to (3.23) we have

1 < Al max v < A1 ((u - u),X

proving (3.17). The upper bound (3.18) for r2 follows easily.


Proof of Theorem 3.4. We may assume without loss of generality, by rotating M by a

fixed element of SO(2, 2), that yo = (2 0,0). Writing + =(u 1, u2, u3) as in (1.13),

the condition (3.11) is equivalent to

1 ch (~2 r)1 u < .2) (3.24)2 < 2

By Theorem 3.2, u satisfies

JAu = (2(h2 + 1) + k) u t

> (h2 + 1)u 1

(h2 + 1)

2

We take

12

(h2 + 1)

in Theorem 3.5. By (2.3) and (3.24) we may take

ch 2T 1= u - k = -(h 2 + 1)

h2 + I - h2 + 1' -

and the result follows easily.

4. Bcklund Transformation of h-Surfaces

In this section we give a Backlund transformation for h-surfaces in H 3. Our method isinspired by those of Pohlmeyer [Po], Eichenherr and Forger [EF] and others concerningtransformations of non-linear sigma models. Our result differs significantly in that theimmersion of an h-surface is only harmonic if h _ 0.

Let be a simply connected Riemann surface and

Z: - M c H31 (4.1)

a conformal immersion of a spacelike h-surface. Let z be a complex coordinate on and let z be a fixed point in Z.

Lemma 4.1. Define the skew symmetric complex-valued matrix J by

J = (J,) = [(Xzp= - ZlZJA) + (fi: - p/o-l)]. (4.2)

The (first order, linear) initial value problem

ox = ( - e ) aJG, E[0, 2t) (*

c)(Zo) = G - diagonal (1, 1, -1, -1)

125

126 B. Palmer

has a unique solution

ax: 2 - S0(2, 2) (4.3)

on Z.

Proof. The integrability condition for (*) is that

Im ca = 0. (4.4)

Using (*) and its complex conjugate one finds

'z = (1 - e'-) (JG + u-'Jz(G)

= 11 - e1 2 a"JGJ + (1 - ei x) rAJfG . (4.5)

If we have

Re J. = 0, (4.6)

then (4.4) reduces to

2(1 - cos A) cr Im JGJ + (1 cos 2) a- Im JG = 0

so that it suffices to show (4.6) and

Im {J, + 2JGJ} = 0. (4.7)

Using (2.5) and (2.6) it follows that

h eeazJal =- (±xa- z Xaz + 2 (X.1- Xp'1.)

h ee

-('1a2±1ip -1az182) + 2 ("1XF 18X) -

so that (4.6) holds and

Im (Jtp)f = 2 Im (-X.fXpz - /2qFZ)

On the other hand

(JGJ)e = { fGyy[(XaXy2 - xyxz) + (tjey2 - v1,7

x [(X,XZ - XpXyz) + (yr1 jz - 14f1ylzl

= XaXz,z<X, x) - XXz<Xf, Xz> - XzXzXz, X> + XzXf(Xz, Xz>

+ XZXp.XZf 1> - XZ1/<XZ, z> + X.z#Xqz, > - XiK7.n<', Xz>+ &01in>10 - qa'in'a<, qlz> - 1aizpl, 0> + I.A2p<q12 ?7z>

ee h e h e h eq e-e

+Xa 2 + XXp + XX + Xp2 2

+ ?l/plz.


Clearly the first, third, fourth and fifth terms above are real valued so that

Im (JGJ),p = Im (zZXpz + ZZrqz)

and (4.4) follows.We next show a'(z) E S0(2, 2) for all z. Suppressing the superscript 1 and denoting

transposition by "tilde" this will follow from

aGd = G. (4.8)

Compute

LaG~d = aGd + aGd-

= const (JGG6 + G(aJG))

= const (J + aGGJd)

=O

since G = G, G2 = id. and J = -J. It follows that G is a constant matrix. Using theinitial condition a(z) = G, the result follows.

Theorem 4.1. There exists a one parameter family of isometric h-surfaces

Z;: Mi c H, [0, 2),

given by

ZX(x) = ;Z

with ca as in the lemma.

Proof.

Xz = azX + a z

= (1 - e) caJGZ + ;XZ

by (*). Since

(JGy), = _ - [(z - XJz) + (pz - lpfz)1 GpZpz

(4.5) gives

Zx = eiA (i.The immersions XA are therefore clearly isomorphic. Next compute, using (*) and (2.5),

z = e (1 - e i) OJGz + aA (e - - ]- ii) X + h )+ h

ee e- aX - h aq.

2 2

127

128 B. Palmer

It follows that the immersions Xi have constant mean curvature h.

Remark. A result similar to the theorem above can be derived for surfaces in S3, H3, or S,.

References

[EF] EICHENHERR, J.; FORGER, M.: On the dual symmetry of the non-linear sigma models.Nuclear Physics B 155 (1979), 381-393.

[G] GAGE, M.: Upper bounds for the first Dirichlet eigenvalue of the Laplace-BeltramiOperator, Indiana U. Math. J., 29, no. 6 (1980), 897-912.

[HO] HOFFMAN, D. H.; OSSERMAN, R.: The Gauss map of surfaces in R3 and R4. Proc.London Math. Soc. (3) 50 (1985), 29-56.

[HOS] HOFFMAN, D. H.; OSSERMAN, R.; SCHOEN, R.: On the Gauss map of complete surfacesof constant mean curvature in R3 and R4. Comment. Math. Helv. 57 (1982), 519-531.

[P] PALMER, B.: Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms.Ann. Global Anal. Geom. 8 (1990), 217-226.

[Po] POHLMEYER, K.: Integrable Hamiltonian systems and interactions through quadraticconstraints. Comm. Math. Phys. 46 (1976), 207-221.

[W] WOLF, J. A.: Spaces of Constant Curvature. Publish or Perish Inc., Boston, Ma., 1972.

BENNETT PALMERTechnische Universitdt BerlinFachbereich 3 - MathematikStr. d. 17. Juni 136W-1000 Berlin 12Bundesrepublik Deutschland

(Received January 25, 1990)

surfaces in lorentzian hyperbolic space

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