Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

TL;DR

This paper studies the evolution of non-compact space-like Cauchy hypersurfaces under prescribed mean curvature flow in certain Lorentzian manifolds, proving long-time existence, preservation of space-likeness, and convergence results.

Contribution

It extends previous results by considering non-compact hypersurfaces with bounded geometry in generalized Robertson-Walker space-times, including convergence analysis.

Findings

Flow preserves space-likeness condition.

Existence of the flow for infinite time.

Convergence results in manifolds with boundary.

Abstract

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.