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

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.

30 pages. This paper sheds light on the relations with the paper arXiv:arch-ive/2202.02424 due to G. Gentile and B. Vertman. They study the prescribed mean curvature flow of non-compact spacelike hypersurfaces in a Generalized Robertson-Walker spacetime (GRW), whereas we study the mean curvature flow of compact hypersurfaces in a GRW with a perpendicular Neumann boundary condition