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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.13152 |
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| _version_ | 1866910671481864192 |
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| author | Chai, Xiaoxiang Wang, Gaoming |
| author_facet | Chai, Xiaoxiang Wang, Gaoming |
| contents | Let $(M, g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\geq 0$. The boundary $\partial M$ is diffeomorphic to the boundary of a rotationally symmetric and weakly convex body $\bar{M}$ in $\mathbb{R}^3$. We call $(\bar{M}, δ)$ a model or a reference. Let $H_{\partial M}$ and $\bar{H}_{\partial M}$ be respectively the mean curvatures of $\partial M$ in $(M, g)$ and $\partial M$ in $(\bar{M}, δ)$, $σ$ and $\barσ$ be the induced metric from $g$ and $δ$. We show that for some classes of $\partial M$, if $H_{\partial M} \geq \bar{H}_{\partial M}$, $σ\geq \barσ$ and the dihedral angles at the nonsmooth part of $\partial M$ are no greater than the model, then $M$ is flat. We also generalize this result to the hyperbolic case and some spaces with $\mathbb{S}^1$-symmetry. Our approach is inspired by Gromov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13152 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scalar curvature comparison of rotationally symmetric sets Chai, Xiaoxiang Wang, Gaoming Differential Geometry 53C42 Let $(M, g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\geq 0$. The boundary $\partial M$ is diffeomorphic to the boundary of a rotationally symmetric and weakly convex body $\bar{M}$ in $\mathbb{R}^3$. We call $(\bar{M}, δ)$ a model or a reference. Let $H_{\partial M}$ and $\bar{H}_{\partial M}$ be respectively the mean curvatures of $\partial M$ in $(M, g)$ and $\partial M$ in $(\bar{M}, δ)$, $σ$ and $\barσ$ be the induced metric from $g$ and $δ$. We show that for some classes of $\partial M$, if $H_{\partial M} \geq \bar{H}_{\partial M}$, $σ\geq \barσ$ and the dihedral angles at the nonsmooth part of $\partial M$ are no greater than the model, then $M$ is flat. We also generalize this result to the hyperbolic case and some spaces with $\mathbb{S}^1$-symmetry. Our approach is inspired by Gromov. |
| title | Scalar curvature comparison of rotationally symmetric sets |
| topic | Differential Geometry 53C42 |
| url | https://arxiv.org/abs/2304.13152 |