Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | |
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| _version_ | 1866908504776769536 |
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| author | Wei, Yong Yang, Bo Zhou, Tailong |
| author_facet | Wei, Yong Yang, Bo Zhou, Tailong |
| contents | We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $α$ of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time $t\in [0,\infty)$. Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06035 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space Wei, Yong Yang, Bo Zhou, Tailong Differential Geometry 53C44, 53C42 We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $α$ of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time $t\in [0,\infty)$. Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data. |
| title | Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space |
| topic | Differential Geometry 53C44, 53C42 |
| url | https://arxiv.org/abs/2210.06035 |