Preprint
Inserted: 1 dec 2020
Last Updated: 1 dec 2020
Year: 2020
Abstract:
We show that any isometric immersion of a flat plane domain into $\R^3$ is developable provided it enjoys $C^{1,\alpha}_{loc}$ H\"older regularity for some $\alpha>2/3$. The proof is based on showing 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 Monge-Amp\`ere equation which appeared in \cite{lepamonge}.
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