Spacelike Hypersurfaces with Constant Scalar Curvature in the Lorentz–Minkowski Space

Abstract

We show that the only compact spacelike hypersurfaces in the Lorentz–Minkowski space Ln+1 having nonzero constant scalar curvature and spherical boundary are the hyperbolic caps (with negative constant scalar curvature). One key ingredient in our proof will be an integral formula for the n-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a hyperplane of Ln+1. As a direct application of that integral formula we also derive an interesting result for the volume of spacelike hypersurfaces.