Stable Surfaces and Free Boundary Marginally Outer Trapped Surfaces

TL;DR

This paper investigates stability notions for surfaces in spacetimes, deriving inequalities, topological results, and estimates for free boundary marginally outer trapped surfaces, extending classical minimal surface results to a relativistic setting.

Contribution

It introduces new stability concepts for MOTS, proves inequalities and topological theorems, and generalizes known minimal surface results to free boundary MOTS in spacetime contexts.

Findings

Proved Christodoulou-Yau estimates for H-stable surfaces.

Established a Cohn-Vossen inequality for non-compact stable MOTS.

Derived area and diameter estimates for free boundary MOTS.

Abstract

We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou-Yau estimate for $\mathbf{H}$-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of $\mathbf{H}$-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively.