Instability of marginally outer trapped surfaces from initial data set symmetry

TL;DR

This paper investigates the stability properties of marginally outer trapped surfaces (MOTS) within initial data sets exhibiting symmetry, revealing how symmetry vector decomposition influences their instability and stability characteristics.

Contribution

It introduces a framework to analyze MOTS stability using symmetry vector decomposition and explores conditions under which instability arises, especially in Einstein manifolds.

Findings

Instability characterized by zero set of normal component

Divergence of symmetry component affects stability

Results specialized for constant mean curvature surfaces

Abstract

Let $(\tilde{\Sigma},h_{ab},K_{ab})$ be an initial data set and let $x^a$ be a symmetry vector of $\tilde{\Sigma}$. Consider a MOTS $\mathcal{S}$ in $\tilde{\Sigma}$ and let the symmetry vector be decomposable along the unit normal to $\mathcal{S}$ in $\tilde{\Sigma}$, and along $\mathcal{S}$. In this note we present some basic results with regards to the stability of $\mathcal{S}$. The vector decomposition allows us to characterize the instability of $\mathcal{S}$ by the nature of the zero set of the normal component to $\mathcal{S}$ and the divergence of the component along $\mathcal{S}$. Further observations are made under the assumption of $\mathcal{S}$ having a constant mean curvature, and $\tilde{\Sigma}$ being an Einstein manifold.