The geometry of $C^{1,\alpha}$ flat isometric immersions

TL;DR

This paper proves that flat isometric immersions with certain regularity are developable surfaces, extending classical results by analyzing weak second fundamental forms and related PDEs.

Contribution

It introduces a weak notion of second fundamental form for $C^{1,rac{2}{3}}$ immersions and proves developability under these regularity conditions.

Findings

Isometric immersions with $c^{1,2/3}$ regularity are developable.

Weak second fundamental form exists for these immersions.

Connections to weak solutions of degenerate Monge-Ampère equations.

Abstract

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little H\"older regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha}$ regularity with $\alpha > 2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge-Amp\`ere equation analyzed by Lewicka and the second author.