Rigidity of Marginally Outer Trapped (Hyper)Surfaces with Negative $\sigma$-Constant

TL;DR

This paper extends rigidity results for marginally outer trapped surfaces (MOTS) with negative sigma-constant, showing that stable, area- or volume-saturating MOTSs imply a splitting of the ambient manifold in various dimensions.

Contribution

It generalizes previous results to MOTSs of higher genus and dimension with negative sigma-constant, providing new splitting theorems in non-time-symmetric initial data sets.

Findings

Stable MOTSs with area or volume bounds lead to manifold splitting.

Results apply to higher genus and high-dimensional MOTSs.

Extends previous theorems to more general initial data sets.

Abstract

In this paper we generalize the main result of [13] in two different situations: in the first case for MOTSs of genus greater than one and, in the second case, for MOTSs of high dimension with negative $\sigma$-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2) or for the volume (in dimension $\ge3$). These results are extensions of [21, Theorem 3] and [20, Theorem 3] to general (non-time-symmetric) initial data sets.