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

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 $ Σ$ is exactly $ -\frac{1}{\sqrtλ}\log τ$, where $ τ$ is the distance from $ Σ$ and $ λ$ is the principal eigenvalue of the MOTS stability operator of $ Σ$. We also prove that the gradient of the solution is of order $ τ^{-1} $. Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.