Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917975317020672 |
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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 $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\infty)$ and converges to a geodesic sphere exponentially as $t\to\infty$ in the smooth topology. A key step is to show the $L^1$ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space Wei, Yong Yang, Bo Zhou, Tailong Differential Geometry Analysis of PDEs 53E10, 53C42 We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\infty)$ and converges to a geodesic sphere exponentially as $t\to\infty$ in the smooth topology. A key step is to show the $L^1$ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence. |
| title | Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space |
| topic | Differential Geometry Analysis of PDEs 53E10, 53C42 |
| url | https://arxiv.org/abs/2406.05159 |