The geometry of C1,α 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 regularity $c^{1,2/3}$ . 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 analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].