A two-dimensional $C^{2,1}$ metric with no local $C^2$ embedding in $\mathbb{R}^3$, following Pogorelov

Abstract

This article presents a proof of Pogorelov's result that there exists a $C^{2,1}$ metric with no local $C^2$ realization in $\mathbb{R}^3$. It also construct in a very elementary way a $C^{1,1}$ realization of this metric. Pogorelov's result is somewhat controversial among the community of researchers that study isometric immersions. This in part owes to the lack of details in Pogorelov's original paper. The chief aim of the paper is therefore to provide the missing details. The construction is the same as Pogorelov's, although the verification differs in some important respects.