Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes

TL;DR

This paper investigates the uniqueness of compact spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, establishing conditions under which they must be spacelike slices.

Contribution

It proves that under the convergence condition, such hypersurfaces are uniquely spacelike slices, using Newton transformations and Minkowski formulae.

Findings

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature is established.

The convergence condition ensures hypersurfaces are spacelike slices.

Methodology involves Newton transformations and Minkowski formulae.

Abstract

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, esentially, under the so called convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.