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

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$isdevelopableprovideditenjoysthelittleH\ddot{o}lderregulairty$c^{1,2/3}$\mathrm{.}Inparticular,isometricimmersionsoflocal$C^{1,\alpha}$regularitywith$\alpha > 2/3$ 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.