Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916740844224512 |
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| author | Santilli, Mario Valentini, Paolo |
| author_facet | Santilli, Mario Valentini, Paolo |
| contents | In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces Santilli, Mario Valentini, Paolo Differential Geometry 53C24, 53C65, 49Q20 In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces. |
| title | Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces |
| topic | Differential Geometry 53C24, 53C65, 49Q20 |
| url | https://arxiv.org/abs/2409.01061 |