Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications

Abstract

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their r-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their r-mean curvatures are linearly related or one of them is constant.

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

“…The general Minkowski formulae for Σ, as derived in [2], are just a simple consequence of our formula (18) and the expression (8). In fact, it follows from (8) and (18) that…”

“…The reason is that it involves covariant derivatives of the Ricci tensor of the ambient space, and one need to assume at least that the ambient space is Einstein to get some control of them. In [2] the authors, jointly with Brasil Jr., developed another method to obtain general Minkowski formulae for spacelike hypersurfaces in conformally stationary spacetimes. Although more analytic than geometric, this method, which follows the ideas of Reilly in [31], has the advantage of working for succesive Minkowski formulae.…”

“…Moreover, the study of such hypersurfaces is also of interest from a physical point of view, because of its relation to several problems in general relativity (see, for instance, [18,19,26]). A basic question on this subject is the uniqueness of spacelike hypersurfaces with constant mean curvature in certain spacetimes (we refer the reader to the introductions of the papers [4] and [2] for a brief account of it). More recently, there has been also an increasing interest in the study of uniqueness of spacelike hypersurfaces with constant higher order mean curvature, including the case of the second order mean curvature which is closely related to the intrinsic scalar curvature of the hypersurface (see Section 2).…”

“…More recently, the authors, jointly with Brasil Jr., initiated the study of hypersurfaces with constant higher order mean curvature in CS spacetimes [2].…”

“…That makes the task much harder, unless one assumes that the ambient spacetime is Einstein or, even more, it has constant sectional curvature. That is the case, for instance, in [1] where higher order Minkowski formulae are developed for spacelike hypersurfaces in de Sitter, in [28] where the third Minkowski formula is written for spacelike hypersurfaces in a CS spacetime with constant sectional curvature, and, more generally, in [2] where the authors, jointly with Brasil Jr., derived general Minkowski formulae for spacelike hypersurfaces in CS spacetimes with constant sectional curvature.…”