The geometry of $C^{1,α}$ flat isometric immersions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866929303130734592 |
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| author | De Lellis, Camillo Pakzad, Mohammad Reza |
| author_facet | De Lellis, Camillo Pakzad, Mohammad Reza |
| contents | 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 $C^{1,α}$ regularity with $α> 2/3$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_11000 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The geometry of $C^{1,α}$ flat isometric immersions De Lellis, Camillo Pakzad, Mohammad Reza Analysis of PDEs Differential Geometry 53C24, 53C21, 57N35, 53A05 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 $C^{1,α}$ regularity with $α> 2/3$ 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. |
| title | The geometry of $C^{1,α}$ flat isometric immersions |
| topic | Analysis of PDEs Differential Geometry 53C24, 53C21, 57N35, 53A05 |
| url | https://arxiv.org/abs/2001.11000 |