Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

TL;DR

This paper establishes long-term existence and geometric properties of mean curvature flow for graphs in Generalized Robertson-Walker spacetimes with Neumann boundary conditions, under null convergence assumptions.

Contribution

It proves long-time existence, conformality of the metric, and preservation of mean convexity for the flow in GRW spacetimes with boundary conditions.

Findings

Longtime existence of the flow in GRW spacetimes.

Conformal relation of the metric to the leaf in asymptotic time.

Preservation of mean convexity during the flow.

Abstract

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.