Hypersurfaces of prescribed scalar curvature in Lorentzian m(4)

rs;

+2F kl¯Rαβγδxαm xβi xγ

k

xδr h m l g rj

?F kl¯Rαβγδxαm xβ

k

xγr xδl h m i g rj?F kl¯Rαβγδxαm xβ

k

xγi xδl h mj

?F kl¯Rαβγδναxβ

k

νγxδl h j i+f¯Rαβγδναxβiνγxδm g mj

+F kl¯Rαβγδ;?{ναxβ

k

xγ

l

xδi x?m g mj+ναxβi xγ

k

xδm x?l g mj}.

The proof is identical to that of the corresponding result in the Riemannian case,cf.[9,Lemma7.1and Lemma7.2];the only di?erence is that f now also depends onν.

Remark3.7.In view of the maximum principle,we immediately deduce from (3.14)that the term(F?f)has a sign during the evolution if it has one at the beginning,i.e.,if the starting hypersurface M0is the upper barrier M2,then (F?f)is non-negative

(3.16)F≥f.

4.Lower order estimates for the auxiliary solutions

Since the two boundary components M1,M2of??are space-like,achronal hypersurfaces,they can be written as graphs over the Cauchy hypersurface S0, M i=graph u i,i=1,2,and we have

(4.1)u1≤u2,

for M1should lie in the past of M2,and the enclosed domain is supposed to be connected.

Let us look at the evolution equation(3.9)with initial hypersurface M0 equal to M2de?ned on a maximal time interval I=[0,T?),T?≤∞.Since the initial hypersurface is a graph over S0,we can write

(4.2)M(t)=graph u(t)|

S0

?t∈I,

The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.

PRESCRIBED SCALAR CUR V ATURE18 where u is de?ned in the cylinder Q T?=I×S0.We then deduce from(3.9), looking at the componentα=0,that u satis?es a parabolic equation of the form

(4.3)˙u=?e?ψv?1(F?f),

where we use the notations in Section2,and where we emphasize that the time derivative is a total derivative,i.e.

?u

(4.4)˙u=

=?e?ψv(F?f).

?t

Thus,?u

The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.

PRESCRIBED SCALAR CUR V ATURE19 The proof uses the relation

(4.8)?v=ηανα

and is identical to that of[15,Lemma4.4]having in mind that presently f also depends onν.

Lemma4.3.Let M(t)=graph u(t)be the?ow hypersurfaces,then,we have

(4.9)˙u?F ij u ij=e?ψ?v f+¯Γ000F ij u i u j

+2F ij¯Γ00i u j+F ij¯Γ0ij,

where all covariant derivatives a taken with respect to the induced metric of the?ow hypersurfaces,and the time derivative˙u is the total time derivative, i.e.it is given by(4.4).

Proof.We use the relation(4.3)together with(2.17).q.e.d.

As an immediate consequence we obtain

Lemma4.4.The composite function

(4.10)?=eµeλu

whereµ,λare constants,satis?es the equation

(4.11)˙??F ij?ij=fe?ψ?vµλeλu?+F ij u i u j¯Γ000µλeλu?

+2F ij u i¯Γ00jµλeλu?+F ij¯Γ0ijµλeλu?

?[1+µeλu]F ij u i u jµλ2eλu?.

Before we can prove the C1-estimates we need two more lemmata. Lemma4.5.There is a constant c=c(?)such that for any positive function 0<?=?(x)on S0and any hypersurface M(t)of the?ow we have

|||ν|||≤c?v,

(4.12)

g ij≤c?v2σij,

(4.13)

F ij≤F kl g kl g ij,

(4.14)