Hypersurfaces of prescribed scalar curvature in Lorentzian m(2)

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 lev …… 此处隐藏：5276字，全部文档内容请下载后查看。喜欢就下载吧 ……