On blow-up solutions of the Jang equation in spherical symmetry

TL;DR

This paper investigates the conditions under which solutions to the Jang equation blow up at various types of marginally outer trapped surfaces in spherical symmetry, revealing new existence results and counterexamples related to the geometry of these surfaces.

Contribution

It extends understanding of blow-up solutions of the Jang equation to include non-outermost MOTSs and provides counterexamples to a previously known geometric property in the non-time-symmetric case.

Findings

Existence of blow-up solutions at non-outermost MOTSs in spherical symmetry.

Not all inner MOTSs admit blow-up solutions, even in spherical symmetry.

Counterexample showing outer-area-minimizing property fails without time-symmetry.

Abstract

We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) \Sigma, there exists a solution to Jang's equation which blows up at \Sigma. Here we show that in addition, large classes of spherically symmetric initial data have solutions to the Jang equation which blow up at non-outermost MOTSs, i.e. MOTSs which lie strictly inside of other MOTS, and even inside of strictly outer trapped surfaces. Unlike for outermost MOTSs, however, we show that there do not \emph{always} exist blow-up solutions for inner MOTSs, even in spherical symmetry. Secondly, an unpublished result of R. Schoen, whose proof we include here, says that in the time-symmetric case, any MOTS corresponding to a blow-up solution for Jang's equation must be outer-area-minimizing, i.e. cannot be contained in a surface of strictly smaller area. The statement is false without the assumption of time-symmetry, however; we construct an explicit spherically symmetric data set providing a counterexample for the general case.