Null mean curvature flow and outermost MOTS

TL;DR

This paper develops a weak solution theory for null mean curvature flow in spacetime initial data, demonstrating convergence to a generalized MOTS and providing insights into the behavior of hypersurfaces near black hole boundaries.

Contribution

It introduces a novel level-set approach for null mean curvature flow and proves the existence and convergence of weak solutions to a generalized MOTS.

Findings

Weak solutions exist for null mean curvature flow starting from mean convex hypersurfaces.

Approximate solutions blow up on the outermost MOTS, indicating a boundary behavior.

Weak solutions converge to a generalized MOTS as limits of finite perimeter sets.

Abstract

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partial\Omega_0$, we show that there exists a weak solution to the null mean curvature flow, given as a limit of approximate solutions that are defined using the $\varepsilon$-regularization method. We show that the approximate solutions blow up on the outermost MOTS and the weak solution converges (as boundaries of finite perimeter sets) to a generalized MOTS.