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

TL;DR

This paper proves Pogorelov's theorem that a specific $C^{2,1}$ metric cannot be locally realized as a $C^2$ surface in $ ^3$, and provides an elementary construction of a $C^{1,1}$ realization, clarifying previous ambiguities.

Contribution

It offers a detailed proof of Pogorelov's non-embeddability result and constructs an explicit $C^{1,1}$ realization, addressing gaps in the original work.

Findings

Existence of a $C^{2,1}$ metric with no local $C^2$ embedding in $ ^3$

Elementary construction of a $C^{1,1}$ realization of this metric

Clarification of Pogorelov's original proof and its details

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.