Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds

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

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HYPERSURF ACES OF PRESCRIBED SCALAR CUR V ATURE IN LORENTZIAN MANIFOLDS CLAUS GERHARDT Dedicated to Robert Finn on the occasion of his eightieth birthday Abstract.The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.Contents 0.Introduction 11.Curvature functions 42.Notations and preliminary results 83.An auxiliary curvature problem 144.Lower order estimates for the auxiliary solutions 175.C 2-estimates for the auxiliary solutions 256.Convergence to a stationary solution 267.Stationary approximations 288.C 1-estimates for the stationary approximations 319.C 2-estimates for the stationary approximations 3610.Existence of a solution 48References 490.Introduction Consider the problem of ?nding a closed hypersurface of prescribed cur-vature F in a globally hyperbolic (n+1)-dimensional Lorentzian manifold N having a compact Cauchy hypersurface S 0.To be more precise,let ?be a connected open subset of N,f ∈C 2,α(¯?),F a smooth,symmetric function de?ned in an open cone Γ?R n ,then we look for a space-like hypersurface M ??such that (0.1)F |M =f (x )?x ∈M,

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

PRESCRIBED SCALAR CUR V ATURE2

where F|

M means that F is evaluated at the vector(κi(x))the components

of which are the principal curvatures of M.The prescribed function f should satisfy natural structural conditions,e.g.ifΓis the positive cone and the hypersurface M is supposed to be convex,then f should be positive,but no further,merely technical,conditions should be imposed.

In[1,2,8,14]the case F=H,the mean curvature,has been treated,and in [15]we solved the problem for curvature functions F of class K?that includes the Gaussian curvature,see[15,Section1]for the de?nition,but excludes the symmetric polynomials H k for1<k<n.Among these,H2,that corresponds to the scalar curvature operator,is of special interest.

However,a solution of equation(0.1)with F=H2is in general not a hy-persurface of prescribed scalar curvature—unless the ambient space has con-stant curvature—since the scalar curvature of a hypersurface also depends on ¯R

αβ

νaνβ.Thus,we have to allow that the right-hand side f also depends on time-like vectors and look for hypersurfaces M satisfying

(0.2)F|

M

=f(x,ν)?x∈M,

whereν=ν(x)is the past-directed normal of M in the point x.

To give a precise statement of the existence result we need a few de?nitions and assumptions.First,we assume that?is a precompact,connected,open subset of N,that is bounded by two achronal,connected,space-like hyper-surfaces M1and M2of class C4,α,where M1is supposed to lie in the past of M2.

Let F=H2be the scalar curvature operator de?ned on the open cone Γ2?R n,and f=f(x,ν)be of class C2,αin its arguments such that

0<c1≤f(x,ν)if ν,ν =?1,

(0.3)

|||fβ(x,ν)|||≤c2(1+|||ν|||2),

(0.4)

and

|||fνβ(x,ν)|||≤c3(1+|||ν|||),

(0.5)

for all x∈¯?and all past directed time-like vectorsν∈T x(?),where|||·|||is a Riemannian reference metric that will be detailed in Section2.

We suppose that the boundary components M i act as barriers for(F,f).

De?nition0.1.M2is an upper barrier for(F,f),if M2is admissible,i.e.its principal curvatures(κi)with respect to the past directed normal belong to Γ2,and if

(0.6)F|

M2

≥f(x,ν)?x∈M2.

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

PRESCRIBED SCALAR CUR V ATURE3 M1is a lower barrier for(F,f),if at the pointsΣ?M1,where M1is admissible,there holds

≤f(x,ν)?x∈Σ.

(0.7)F|

Σ

Σmay be empty.

Remark0.2.This de?nition of upper and lower barriers for a pair(F,f)also makes sense for other curvature functions F de?ned in an open convex cone Γ,with a corresponding meaning of the notion admissable.

Now,we can state the main theorem.

Theorem0.3.Let M1be a lower and M2an upper barrier for(F,f),where F=H2.Then,the problem

=f(x,ν)

(0.8)F|

M

has an admissible solution M?¯?of class C4,αthat can be written as a graph over S0provided there exists a strictly convex functionχ∈C2(¯?).

Remark0.4.As we have shown in[15,Lemma2.7]the existence of a strictly convex functionχis guaranteed by the assumption that the level hypersurfaces {x0=const}are strictly convex in¯?,where(xα)is a Gaussian coordinate system associated with S0.

Looking at Robertson-Walker space-times it seems that the assumption of the existence of a strictly convex function in the neighbourhood of a given compact set is not too restrictive:in Minkowski space e.g.χ=?|x0|2+|x|2is a globally de?ned strictly convex function.The only obstruction we are aware of is the existence of a compact maximal slice.In the neighbourhood of such a slice a strictly convex function cannot exist.

The existence result of our main theorem would also be valid in Riemannian manifolds if one could prove C1-estimates.For the C2-estimates the nature of the ambient space is irrelevant though the proofs are slightly di?erent.

For prescribed curvature problems it seems more natural to assume that the right-hand side f depends on(x,ν),and we shall prove in a subsequent paper existence results for curvature functions F∈(K?),where the ambient space can be Riemannian or Lorentzian,cf.[16].

The paper is organized as follows:In Section1we take a closer look at curvature functions and de?ne the concept of elliptic regularization for these functions,and analyze some of its properties.

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

PRESCRIBED SCALAR CUR V ATURE4 In Section2we introduce the notations and common de?nitions we rely on,and state the equations of Gauß,Codazzi,and Weingarten for space-like hypersurfaces.

In Section3we look at the curvature?ow associated with our problem,and the corresponding evolution equations for the basic geometrical quantities of the?ow hypersurfaces.

In Section4we prove lower order estimates for the evolution problem,while a priori estimates in the C2-norm are derived in Section5.

In Section6,we demonstrate that the evolutionary solution converges to a stationary approximation of our problem,i.e.to a solution for a curvature problem,where F is replaced by its elliptic regularization F?.

The uniform C1-estimates for the stationary approximations are derived in Sections7and8,the C2-estimates are given in Section9,while the?nal existence result is contained in Section10.

1.Curvature functions

LetΓ?R n be an open cone containing the positive coneΓ+,and F∈C2,α(Γ)∩C0(¯Γ)a positive symmetric function satisfying the condition

?F

(1.1)F i=

(1.3)

?h ij

and

?2F

F ij,kl=

|ξi|2?ξ∈R n,

?κi

in an appropriate coordinate system,

(1.6)F ij is diagonal if h ij is diagonal,

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

PRESCRIBED SCALAR CUR V ATURE5 and

(1.7)F ij,klηijηkl=?2F

κi?κj

(ηij)2,

for any(ηij)∈S,where S is the space of all symmetric matrices.The second term on the right-hand side of(1.7)is non-positive if F is concave,and non-negative if F is convex,and has to be interpreted as a limit ifκi=κj.

The preceding considerations are also applicable if theκi are the principal curvatures of a space-like hypersurface M with metric(g ij).F can then be looked at as being de?ned on the space of all symmetric tensors(h ij)the eigen-values of which belong toΓ.Such tensors will be called admissible;when the second fundamental form of M is admissible,then,we also call M admissible.

For an admissible tensor(h ij)

(1.8)F ij=

?F

?h i j

is also a mixed tensor with contravariant index j and covariant index i.

Such functions F are called curvature functions.Important examples are the symmetric polynomials of order k,H k,1≤k≤n,

(1.11)H k(κi)= i1<···<i kκi1···κi k.

They are de?ned on an open coneΓk that can be characterized as the connected component of{H k>0}that containsΓ+.

Since we have in mind that theκi are the principal curvatures of a hyper-surface,we use the standard symbols H and|A|for

H= iκi,

(1.12)

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

PRESCRIBED SCALAR CUR V ATURE 6

and

|A |2=

i κ2i .

(1.13)The scalar curvature function F =H 2can then be expressed as (1.14)F =1

2H 2+1

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

PRESCRIBED SCALAR CUR V ATURE7 Then,

(1.23)

Γ?Γ?,

and

(1.24)

H>0inΓ?.

Proof.We only prove the assertions(1.23)and(1.24)since the other assertions are obvious.Let(κi)∈Γbe?xed.Then,

(1.25)0<F(κi)≤F(κi+?H),

because F is monotone,and we deduce

(1.26)(κi+?H)∈Γ??>0,

in view of(1.22)and the monotonicity of F,cf.(1.1).

To prove(1.24),we observe that

(1.27) i?κi=(1+?n) iκi.

q.e.d.

Remark1.2.(i)Let F be as in Lemma1.1and assume moreover,that F is homogeneous of degree1,and concave,then,

1

(1.28)F(κi)≤

d0concave,then,the relation(1.22)is also valid.

Proof.The inequality(1.28)follows easily from the concavity and homogeneity

F(κi)≤F(1,...,1)+ i F i(1,...,1)(κi?1)

(1.29)

1

=

F(1,...,1),while the other assertions are obvious.q.e.d.

n

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

PRESCRIBED SCALAR CUR V ATURE8 For better reference,we use a tensor setting in the next lemma,i.e.the (κi)∈Γare the eigenvalues of an admissible tensor(h ij)with respect to a Riemannian metric(g ij).In this setting the elliptic regularization of F is given by

(1.30)?F(h ij)≡F(h ij+?Hg ij).

Lemma1.3.Let?F be the elliptic regularization of a curvature function F of class C2,then,

(1.31)?F ij=F ij+?F rs g rs g ij,

and

(1.32)?F ij,kl=F ij,kl+?F ij,ab g

ab

g kl

+?F rs,kl g rs g ij+?2F rs,ab g rs g ab g ij g kl.

If F is concave,then,?F is also concave.

Proof.The relations(1.31)and(1.32)are straight-forward calculations.

To prove the concavity of?F,let(ηij)be a symmetric tensor,then,

(1.33)?F ij,klη

ij

ηkl=F ij,klηijηkl+2?F ij,rsηij g rs g klηkl

+?2F rs,ab g rs g ab(g ijηij)2≤0.

q.e.d.

2.Notations and preliminary results

The main objective of this section is to state the equations of Gauß,Co-dazzi,and Weingarten for space-like hypersurfaces M in a(n+1)-dimensional Lorentzian space N.Geometric quantities in N will be denoted by(¯gαβ),(¯Rαβγδ), etc.,and those in M by(g ij),(R ijkl),etc.Greek indices range from0to n and Latin from1to n;the summation convention is always used.Generic co-ordinate systems in N resp.M will be denoted by(xα)resp.(ξi).Covariant di?erentiation will simply be indicated by indices,only in case of possible am-biguity they will be preceded by a semicolon,i.e.for a function u in N,(uα)

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

PRESCRIBED SCALAR CUR V ATURE9 will be the gradient and(uαβ)the Hessian,but e.g.,the covariant derivative of the curvature tensor will be abbreviated by¯Rαβγδ;?.We also point out that

(2.1)¯Rαβγδ;i=¯Rαβγδ;?x?i

with obvious generalizations to other quantities.

Let M be a space-like hypersurface,i.e.the induced metric is Riemannian, with a di?erentiable normalνthat is time-like.

In local coordinates,(xα)and(ξi),the geometric quantities of the space-like hypersurface M are connected through the following equations

(2.2)xαij=h ijνα

the so-called Gaußformula.Here,and also in the sequel,a covariant derivative is always a full tensor,i.e.

(2.3)xαij=xα,ij?Γk ij xαk+¯Γαβγxβi xγj.

The comma indicates ordinary partial derivatives.

In this implicit de?nition the second fundamental form(h ij)is taken with respect toν.

The second equation is the Weingarten equation

(2.4)ναi=h k i xαk,

where we remember thatναi is a full tensor.

Finally,we have the Codazzi equation

(2.5)h ij;k?h ik;j=¯Rαβγδναxβi xγj xδk

and the Gaußequation

xδl.

(2.6)R ijkl=?{h ik h jl?h il h jk}+¯Rαβγδxαi xβj xγ

k

Now,let us assume that N is a globally hyperbolic Lorentzian manifold with a compact Cauchy surface.N is then a topological product R×S0,where S0is a compact Riemannian manifold,and there exists a Gaussian coordinate system (xα),such that x0represents the time,the(x i)1≤i≤n are local coordinates for S0,where we may assume that S0is equal to the level hypersurface{x0=0}—we don’t distinguish between S0and{0}×S0—,and such that the Lorentzian metric takes the form

(2.7)d¯s2N=e2ψ{?dx02+σij(x0,x)dx i dx j},

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

PRESCRIBED SCALAR CUR V ATURE10 whereσij is a Riemannian metric,ψa function on N,and x an abbreviation for the space-like components(x i),see[17],[19,p.212],[18,p.252],and [8,Section6].We also assume that the coordinate system is future oriented, i.e.the time coordinate x0increases on future directed curves.Hence,the contravariant time-like vector(ξα)=(1,0,...,0)is future directed as is its covariant version(ξα)=e2ψ(?1,0,...,0).

Let M=graph u|

S0

be a space-like hypersurface

(2.8)M={(x0,x):x0=u(x),x∈S0},

then the induced metric has the form

(2.9)g ij=e2ψ{?u i u j+σij}

whereσij is evaluated at(u,x),and its inverse(g ij)=(g ij)?1can be expressed as

(2.10)g ij=e?2ψ{σij+u i

v

},

where(σij)=(σij)?1and

(2.11)u i=σij u j

v2=1?σij u i u j≡1?|Du|2.

Hence,graph u is space-like if and only if|Du|<1.

We also note that

(2.12)v?2=1+e2ψg ij u i u j≡1+e2ψ Du 2.

The covariant form of a normal vector of a graph looks like (2.13)(να)=±v?1eψ(1,?u i).

and the contravariant version is

(2.14)(να)=?v?1e?ψ(1,u i).

Thus,we have

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

PRESCRIBED SCALAR CUR V ATURE11 Remark2.1.Let M be space-like graph in a future oriented coordinate sys-tem.Then,the contravariant future directed normal vector has the form (2.15)(να)=v?1e?ψ(1,u i)

and the past directed

(2.16)(να)=?v?1e?ψ(1,u i).

In the Gaußformula(2.2)we are free to choose the future or past directed normal,but we stipulate that we always use the past directed normal for reasons that we have explained in[15].

Look at the componentα=0in(2.2)and obtain in view of(2.16)

(2.17)e?ψv?1h ij=?u ij?¯Γ000u i u j?¯Γ00j u i?¯Γ00i u j?¯Γ0ij.

Here,the covariant derivatives are taken with respect to the induced metric of M,and

(2.18)?¯Γ0ij=e?ψ¯h ij,

where(¯h ij)is the second fundamental form of the hypersurfaces{x0=const}.

An easy calculation shows

(2.19)¯h ij e?ψ=?1

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

PRESCRIBED SCALAR CUR V ATURE 12

Remark 2.4.The M i are barriers for the pair (F,f ).Let us point out that without loss of generality we may assume

F |M 2>f (x,ν)?x ∈M 2,

(2.20)

and

F |Σ<f (x,ν)?x ∈Σ,(2.21)for let η∈C ∞(¯?)be a function with support in a small neighbourhood of M 1.∪M 2—the dot should indicate that the union is disjoint—such that

(2.22)

η|M 1>0and η|M 2<0

and de?ne for δ>0

(2.23)f δ=f +δη.Then,if we assume f to be strictly positive with a positive lower bound,we have for small δ

(2.24)f δ≥1

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

PRESCRIBED SCALAR CUR V ATURE13 contradicting(2.21).q.e.d.

Remark2.6.The condition(0.3)is reasonable as is evident from the Einstein equation

(2.27)¯Rαβ?1

≤?and?(t)=t?t≥c1,

2

then,we can replace f by??f and the new function satis?es our requirements for all time-like vectors.

We therefore assume in the following that the relation(0.3)holds for all time-like vectorsν∈T x(N)and all x∈¯?.

Sometimes,we need a Riemannian reference metric,e.g.if we want to estimate tensors.Since the Lorentzian metric can be expressed as

(2.30)¯gαβdxαdxβ=e2ψ{?dx02+σij dx i dx j},

we de?ne a Riemannian reference metric(?gαβ)by

(2.31)?gαβdxαdxβ=e2ψ{dx02+σij dx i dx j}

and we abbreviate the corresponding norm of a vector?eldηby

(2.32)|||η|||=(?gαβηαηβ)1/2,

with similar notations for higher order tensors.

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

PRESCRIBED SCALAR CUR V ATURE14 For a space-like hypersurface M=graph u the induced metrics with respect to(¯gαβ)resp.(?gαβ)are related as follows

(2.33)?g ij=?gαβxαi xβj=e2ψ[u i u j+σij] =g ij+2e2ψu i u j.

Thus,if(ξi)∈T p(M)is a unit vector for(g ij),then

(2.34)?g ijξiξj=1+2e2ψ|u iξi|2,

and we conclude for future reference

Lemma2.7.Let M=graph u be a space-like hypersurface in N,p∈M,and ξ∈T p(M)a unit vector,then

(2.35)|||xβiξi|||≤c(1+|u iξi|)≤c?v,

where?v=v?1.

3.An auxiliary curvature problem

Solving the problem(0.2)involves two steps:?rst,proving a priori esti-mates,and secondly,applying a method to show the existence of a solution. In a general Lorentzian manifold the evolution method is the method of choice, but unfortunately,one cannot prove the necessary a priori estimates during the evolution when F is the scalar curvature operator.Both the C1and C2-esti-mates fail for general f=f(x,ν).

Therefore,we use the elliptic regularization and consider the existence prob-lem for the operators

(3.1)F?(κi)=F(κi+?H),?>0,

i.e.we solve

(3.2)F?|

M

=f(x,ν).

Then,we prove uniform C2,α-estimates for the approximating solutions M?,and?nally,let?tend to zero.

The F?—or some positive power of it—belong to a class of curvature func-tions F that satisfy the following condition(H):F∈C2,α(Γ)∩C0(¯Γ), whereΓ?R n is an open cone containingΓ+,F is symmetric,monotone,

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

PRESCRIBED SCALAR CUR V ATURE15 i.e.F i>0,homogeneous of degree1,concave,vanishes on?Γ,and there exists?0=?0(F)>0such that

(3.3)F i≥?0 k F k?1≤i≤n.

Furthermore,the set

(3.4)Λδ,κ={(κi)∈Γ:0<δ≤F(κi),κi≤κ?1≤i≤n}

is compact.

Remark3.1.If the original curvature function F∈C2,α(Γ)∩C0(¯Γ)is con-cave,homogeneous of degree1,and vanishes on?Γ,then,the F?are of class (H)in the coneΓ?,and satisfy(3.3)with?0=?.The set

(3.5)?Λδ,κ={(κi)∈Γ?:0<δ≤F?(κi),κi≤κ?1≤i≤n}

is compact for?xed?.

If the parametersκandδare independent of?,then the?Λδ,κare contained in a compact subset ofΓuniformly in?,for small?,0≤?≤?1(δ,κ,F). Proof.In view of the results in Lemma1.3we only have to prove the com-pactness of?Λδ,κ.We shall also only consider the case when the estimates hold uniformly in?.

Due to the concavity and homogeneity of F?we conclude from(1.28)that (3.6)F?(κi)≤

1

n F(1,...,1)H≤(1+n?)F(1,...,1)κ,

(3.7) and thus,

lim ?→0?H=0,

(3.8)

uniformly in?Λδ,κ.

Suppose?Λδ,κwould not stay in a compact subset ofΓfor small?,0<?≤?1(δ,κ,F).Then,there would exist a sequence?→0and a corresponding sequence(κ?i)∈?Λδ,κconverging to a point(κi)∈?Γ,which is impossible in view of(3.7),(3.8),and the continuity of F in¯Γ.q.e.d.

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

PRESCRIBED SCALAR CUR V ATURE16 To prove the existence of hypersurfaces of prescribed curvature F for F∈(H)we look at the evolution problem

(3.9)

˙x=(F?f)ν, x(0)=x0,

whereνis the past-directed normal of the?ow hypersurfaces M(t),F the curvature evaluated at M(t),x=x(t)an embedding and x0an embedding of an initial hypersurface M0,which we choose to be the upper barrier M2.

Since F is an elliptic operator,short-time existence,and hence,existence in a maximal time interval[0,T?)is guaranteed.If we are able to prove uniform a priori estimates in C2,α,long-time existence and convergence to a stationary solution will follow immediately.

But before we prove the a priori estimates,we want to show how the metric, the second fundamental form,and the normal vector of the hypersurfaces M(t) evolve.All time derivatives are total derivatives.The proofs are identical to those of the corresponding results in a Riemannian setting,cf.[9,Section3] and[15,Section4],and will be omitted.

Lemma3.2(Evolution of the metric).The metric g ij of M(t)satis?es the evolution equation

(3.10)˙g ij=2(F?f)h ij.

Lemma3.3(Evolution of the normal).The normal vector evolves according to

(3.11)˙ν=?M(F?f)=g ij(F?f)i x j.

Lemma3.4(Evolution of the second fundamental form).The second funda-mental form evolves according to

(3.12)˙h j i=(F?f)j i?(F?f)h k i h j

k

?(F?f)¯Rαβγδναxβiνγxδk g kj

and

(3.13)˙h ij=(F?f)ij+(F?f)h k i h kj?(F?f)¯Rαβγδναxβiνγxδj.

Lemma3.5(Evolution of(F?f)).The term(F?f)evolves according to the equation

(3.14)(F?f)′?F ij(F?f)ij=?F ij h ik h k j(F?f)?fανα(F?f)

?fναxαi(F?f)j g ij?F ij¯Rαβγδναxβiνγxδj(F?f),

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

PRESCRIBED SCALAR CUR V ATURE17 From(3.9)we deduce with the help of the Ricci identities a parabolic equa-tion for the second fundamental form

Lemma3.6.The mixed tensor h j

i

satis?es the parabolic equation

(3.15)˙h j

i

?F kl h j

i;kl

=?F kl h rk h r l h j i+fh k i h j

k

?fαβxαi xβ

k

g kj?fαναh j i?fανβ(xαi xβ

k

h kj+xαl xβ

k

h k i g lj)

?fνανβxαl xβ

k

h k i h lj?fνβxβ

k

h k i;l g lj?fναναh k i h j

k

+F kl,rs h kl;i h j

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)

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