Remarks on the size of apparent horizons

TL;DR

This paper demonstrates that stable apparent horizons in certain initial data sets are not only area-bounded but also satisfy a diameter bound, providing new geometric constraints on these surfaces.

Contribution

The paper introduces the novel result that stable apparent horizons obey a diameter bound, extending previous area bounds in the context of initial data sets with dominant energy condition.

Findings

Stable apparent horizons have a diameter bound

The diameter bound complements known area bounds

Multiple methods are used to establish the diameter bound

Abstract

Marginally outer trapped surfaces (also referred to as apparent horizons) that are stable in 3-dimensional initial data sets obeying the dominant energy condition strictly are known to satisfy an area bound. The main purpose of this note is to show (in several ways) that such surfaces also satisfy a diameter bound.