Blowup rate control for solution of Jang’s equation and its application to Penrose inequality

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.