Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.08198 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929589220016128 |
|---|---|
| author | Fong, Frederick Tsz-Ho |
| author_facet | Fong, Frederick Tsz-Ho |
| contents | In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $-1$ (including the inverse mean curvature flow) in warped product spaces $I \times_ϕ M^n$, where $M^n$ is a compact homogeneous manifold and $ϕ'' \geq 0$, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $-1$ and with an extra assumption $ϕ' \geq 0$.
Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces Fong, Frederick Tsz-Ho Differential Geometry 53E10 In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $-1$ (including the inverse mean curvature flow) in warped product spaces $I \times_ϕ M^n$, where $M^n$ is a compact homogeneous manifold and $ϕ'' \geq 0$, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $-1$ and with an extra assumption $ϕ' \geq 0$. Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric. |
| title | Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces |
| topic | Differential Geometry 53E10 |
| url | https://arxiv.org/abs/2411.08198 |