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

Abstract

We show that any isometric immersion of a flat plane domain into ${\mathbb{R}}^3$ is developable provided it enjoys the little Hölder 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-Ampère equation analyzed by Lewicka and the second author.

31 pages; Minor editorial corrections with respect to v4. To appear in Proceedings of the Royal Society of Edinburgh Section A

References in corpus (1)

• Dissipative continuous Euler flows