#### Abstract

Solutions to the classical Weyl problem by Nirenberg and Pogorelov
play fundamental role in the notion of quasi local masses and positive quasi

local mass theorems in general relativity. An interesting question in

differential geometry is whether one can isometrically embed compact surfaces

with positive Gauss curvature to a general 3 dimensional ambient space. Of

particular importance is the anti de Sitter Schwarzchild space in general

relativity. We discuss some recent progress in this direction, the a priori

estimates for embedded surfaces in a joint work with Lu, the openness and

non-rigidity results of Li -Wang, and a new quasi local type inequality of

Lu-Miao. We will also discuss open problem related to isometric embeddings to

ambient spaces with horizons.