Hypersurfaces of prescribed scalar curvature in Lorentzian m

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 …… 此处隐藏：5564字，全部文档内容请下载后查看。喜欢就下载吧 ……