Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

TL;DR

This paper establishes conditions for the short-term existence, long-term behavior, and convergence of prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds, with applications to Minkowski and Schwarzschild spacetimes.

Contribution

It provides new criteria for the existence and convergence of prescribed mean curvature flow in noncompact Lorentzian settings, extending previous compact or asymptotically flat results.

Findings

Short time existence under certain conditions

Long time existence and convergence results

Applications to Minkowski and Schwarzschild spacetimes

Abstract

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed mean curvature flow on a Lorentz manifold with a smooth time function starting from a complete noncompact spacelike hypersurface. Long time existence and convergence will also be discussed. Results will be applied to study some prescribed mean curvature flows inside the future of the origin in the Minkowski spacetime. Examples of spacetime related to the existence and convergence results near the future null infinity of the Schwarzschild spacetime are also discussed.