Isometric embedding with nonnegative Gauss curvature under the graph setting

Abstract

We study the regularity of the isometric embedding $ X: $ $ (B(O, r), g) $ $ \rightarrow $ $ (\mathbb{R}^3, g_{can}) $ of a 2-ball with nonnegatively curved $ C^4 $ metric into $ \mathbb{R}^3 $. Under the assumption that $ X $ can be expressed in the graph form, we show $ X \in C^{2,1} $ near $ P $, which is optimal by Iaia's example.