Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915150919892992 |
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| author | Meier, Damaris Ntalampekos, Dimitrios |
| author_facet | Meier, Damaris Ntalampekos, Dimitrios |
| contents | We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric surfaces that we establish, and which generalizes a recent result of Esmayli--Ikonen--Rajala. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07621 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces Meier, Damaris Ntalampekos, Dimitrios Metric Geometry Complex Variables Differential Geometry Primary 53C23, 53C45, Secondary 30C65, 53A05 We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric surfaces that we establish, and which generalizes a recent result of Esmayli--Ikonen--Rajala. |
| title | Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces |
| topic | Metric Geometry Complex Variables Differential Geometry Primary 53C23, 53C45, Secondary 30C65, 53A05 |
| url | https://arxiv.org/abs/2305.07621 |