Mean curvature flow in null hypersurfaces and the detection of MOTS

TL;DR

This paper investigates the mean curvature flow within null hypersurfaces in spacetime, demonstrating conditions under which the flow exists globally and converges to a MOTS, aiding in understanding black hole horizons.

Contribution

It introduces a novel analysis of mean curvature flow in null hypersurfaces and proves convergence to MOTS under mild conditions, extending geometric flow techniques to relativistic settings.

Findings

Flow exists for all time under certain conditions.

Flow converges smoothly to a MOTS.

Provides a foliation of the spacetime past of a MOTS.

Abstract

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.