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arXiv · 2001.11000

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

Camillo De Lellis·Mohammad Reza Pakzad
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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.

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Camillo De Lellis, Mohammad Reza Pakzad. 2020-01-29. The geometry of $C^{1,\alpha}$ flat isometric immersions. https://arxiv.org/abs/2001.11000

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