Blowup of Jang's equation at outermost marginally trapped surfaces

TL;DR

This paper studies the blowup behavior of Jang's equation at outermost marginally trapped surfaces, including construction of blowup solutions, exclusion of additional blowup surfaces, and convergence rates near stable surfaces.

Contribution

It provides new insights into the blowup phenomena of Jang's equation, including construction methods, conditions for the absence of extra blowup surfaces, and exponential convergence results.

Findings

Constructed solutions blowing up at outermost MOTS.

Excluded extra blowup surfaces in non-positive mean curvature data.

Proved exponential convergence of blowup near stable MOTS.

Abstract

The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence near a strictly stable MOTS.