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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2111.10170 |
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| _version_ | 1866908989889970176 |
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| author | Hong, Fang |
| author_facet | Hong, Fang |
| contents | In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$ with speed $(\sinh r)^{α/β} σ_{k}^{{1}/β}$, where $σ_{k}$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $α$, $ β$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $α$ and $ β$. When $k = 1 , α> 1 + β$, and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for $ k=1 $ exists for all time and converges smoothly to a sphere. When $1\leq k \leq n, α> k+β$, and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang's results from Euclidean space to hyperbolic space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10170 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space Hong, Fang Differential Geometry In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$ with speed $(\sinh r)^{α/β} σ_{k}^{{1}/β}$, where $σ_{k}$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $α$, $ β$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $α$ and $ β$. When $k = 1 , α> 1 + β$, and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for $ k=1 $ exists for all time and converges smoothly to a sphere. When $1\leq k \leq n, α> k+β$, and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang's results from Euclidean space to hyperbolic space. |
| title | On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2111.10170 |