In this talk, we will first review the classic result of isometric embedding of (S^2,g) into 3-dimensional Euclidean space by Nirenberg and Pogorelov. We will then discuss how to apply it to define quasi-local mass in general relativity. In particular, the positivity of Brown-York quasi-local mass proved by Shi-Tam is equivalent to the Riemannian Positive mass theorem by Schoen-Yau and Witten.

We will then discuss recent progresses in isometric embedding of (S^2,g) into general Riemannian manifold. We will also discuss recent works on a localized Riemannian Penrose inequality, which is equivalent to the Riemannian Penrose inequality.

This is based on joint works with P. Guan and P. Miao. 

1.14.pdf