On blowup of regularized solutions to Jang equation and constant expansion surfaces

TL;DR

This paper investigates the blowup behavior of regularized solutions to the Jang equation within apparent horizons, extending previous analyses and characterizing the limits of solutions through geometric treatments like dilation and translation.

Contribution

It introduces new geometric methods to analyze blowup limits of solutions to the Jang equation inside apparent horizons, including translation and rescaling techniques.

Findings

Limits of translated solutions are constant expansion surfaces.

Rescaled solutions converge to specific geometric structures.

Analysis of blowup regions enclosed by apparent horizons.

Abstract

In this paper, we analyze the blowup behavior of regularized solutions to Jang equation inside apparent horizons. This extends the analyses outside apparent horizons done by Schoen-Yau. We will take two natural geometric treatments to blowup sequences: dilation and translation. First, we show that the limits of properly translated solutions are constant expansion surfaces. Second, we characterize the limits of properly rescaled solutions. Third, we discuss the structure of blowup regions enclosed by apparent horizons. Lastly, we elaborate on a special case of low-speed blowup.