Hypersurfaces of prescribed scalar curvature in Lorentzian m(3)

(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 …… 此处隐藏：5223字，全部文档内容请下载后查看。喜欢就下载吧 ……