Rigidity and Flexibility of Isometric Extensions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866914965951086592 |
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| author | Cao, Wentao Inauen, Dominik |
| author_facet | Cao, Wentao Inauen, Dominik |
| contents | In this paper we consider the rigidity and flexibility of $C^{1, θ}$ isometric extensions and we show that the Hölder exponent $θ_0=\frac12$ is critical in the following sense: if $u\in C^{1,θ}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $Σ$ and $θ> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $Σ$. On the other hand, for any $θ<\frac12$ we can construct $C^{1,θ}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, θ}$ isometric embeddings, $θ<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_00418 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rigidity and Flexibility of Isometric Extensions Cao, Wentao Inauen, Dominik Analysis of PDEs Differential Geometry 53B20, 53A07, 57R40, 35F60, 58B20 In this paper we consider the rigidity and flexibility of $C^{1, θ}$ isometric extensions and we show that the Hölder exponent $θ_0=\frac12$ is critical in the following sense: if $u\in C^{1,θ}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $Σ$ and $θ> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $Σ$. On the other hand, for any $θ<\frac12$ we can construct $C^{1,θ}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, θ}$ isometric embeddings, $θ<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension. |
| title | Rigidity and Flexibility of Isometric Extensions |
| topic | Analysis of PDEs Differential Geometry 53B20, 53A07, 57R40, 35F60, 58B20 |
| url | https://arxiv.org/abs/2010.00418 |