Some rigidity results related to the Obata type equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909513655779328 |
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| author | Liu, Yiwei Yang, Yihu |
| author_facet | Liu, Yiwei Yang, Yihu |
| contents | Let $(Ω^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partialΩ=Σ$ and $f$ a smooth function on $Ω$ which satisfies the Obata type equation $\nabla^2 f -fg =0$ with Robin boundary condition $f_ν = cf$, where $c=\cothθ>1$. In this paper, we provide some rigidity results based on the warped product structure of $Ω$ determined by the equation $\nabla^2 f -fg =0$ and appropriate curvature assumptions. We also apply a similar method to the Obata type equation $\nabla^2 f +fg =0$ and get a rigidity result on the standard sphere $\mathbb{S}^{n+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some rigidity results related to the Obata type equation Liu, Yiwei Yang, Yihu Differential Geometry Let $(Ω^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partialΩ=Σ$ and $f$ a smooth function on $Ω$ which satisfies the Obata type equation $\nabla^2 f -fg =0$ with Robin boundary condition $f_ν = cf$, where $c=\cothθ>1$. In this paper, we provide some rigidity results based on the warped product structure of $Ω$ determined by the equation $\nabla^2 f -fg =0$ and appropriate curvature assumptions. We also apply a similar method to the Obata type equation $\nabla^2 f +fg =0$ and get a rigidity result on the standard sphere $\mathbb{S}^{n+1}$. |
| title | Some rigidity results related to the Obata type equation |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2411.18508 |