Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866914664980414464 |
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| author | Cui, Jingshi Zhao, Peibiao |
| author_facet | Cui, Jingshi Zhao, Peibiao |
| contents | In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $λ^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $Ω$ is the domain enclosed by $M$. In particular, when $f$ is of constant and $k$ is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00855 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows Cui, Jingshi Zhao, Peibiao Differential Geometry In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $λ^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $Ω$ is the domain enclosed by $M$. In particular, when $f$ is of constant and $k$ is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei. |
| title | Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2211.00855 |