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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.07361 |
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| _version_ | 1866916890435125248 |
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| author | Sheng, Weimin Yang, Jiazhuo |
| author_facet | Sheng, Weimin Yang, Jiazhuo |
| contents | In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its speed $f(r)σ_k^α$ where $σ_k$ is the $k$-th elementary symmetric polynomial of $M$'s principle curvatures, $r$ is the distance of the point on $M$ to the origin, $f$ is a smooth nonnegative function on $[0,\infty)$ and $α> 0$. Under some suitable conditions on $f$, we prove that starting from a star-shaped and $k$-convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in \cite{li2022asymptotic}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows Sheng, Weimin Yang, Jiazhuo Differential Geometry In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its speed $f(r)σ_k^α$ where $σ_k$ is the $k$-th elementary symmetric polynomial of $M$'s principle curvatures, $r$ is the distance of the point on $M$ to the origin, $f$ is a smooth nonnegative function on $[0,\infty)$ and $α> 0$. Under some suitable conditions on $f$, we prove that starting from a star-shaped and $k$-convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in \cite{li2022asymptotic}. |
| title | Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.07361 |