Blowup rate control for solution of Jang's equation and its application on Penrose inequality

TL;DR

This paper precisely characterizes the blowup behavior of solutions to Jang's equation near stable MOTS and applies these insights to derive a Penrose-like inequality, advancing understanding in geometric analysis and general relativity.

Contribution

It provides an exact description of the blowup rate of Jang's equation solutions near stable MOTS and applies this to establish a Penrose-like inequality.

Findings

Blowup term is exactly -1/√λ log τ near MOTS

Gradient of solution is of order τ^{-1}

Derived a Penrose-like inequality under certain conditions

Abstract

We prove that the blowup term of a blowup solution of Jang's equation on an initial data set (M,g,k) near an arbitrary strictly stable MOTS $ \Sigma $ is exactly $ -\frac{1}{\sqrt{\lambda}}\log \tau $, where $ \tau $ is the distance from $ \Sigma $ and $ \lambda $ is the principal eigenvalue of the MOTS stability operator of $ \Sigma $. We also prove that the gradient of the solution is of order $ \tau^{-1} $. Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.