R-maximal Spacelike Submanifolds In Lorentz Space Forms

Let x:M→N1n+p(c)be an n-dimensional compact spacelike submanifold in an(n+p)-dimensional Lorentz space form N1n+p(c).Assume that r∈{0,1,…,n-1}and r is even.In this paper,we firstly introduce r-th mean curvature function Sr and(r+1)-th mean curvature vector field Sr+1.If Sr+1 ≡0 on M,we call M to be an r-maximal spacelike submanifold.Secondly,we define a functional Jr(x)=∫MFr(S0,S2,…,ST)dv of x:M→N1n+p(c).By calculation of the first variational formula of Jr,we show that x is a critical point of Jr if and only if x is r-maximal.Furthermore,we give some examples of r-maximal submanifolds in Lorentz space forms.By calculating the second varitional formula of Jr,we prove that there exists no compact without boundary stable r-maximal spacelike hypersurface with positive r-th mean curvature in de Sitter space S1n+1(c).Finally,by investigating complete Willmore maximal spacelike hypersurfaces with constant scalar curvatur in anti-de Sitter space H15(-1),we give a new characterization of hyperbolic cylinder H2(-2)× H2(-2)in H15(-1).