Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929715475906560 |
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| author | Pessoa, Leandro F. Véras, Erisvaldo Vieira, Bruno |
| author_facet | Pessoa, Leandro F. Véras, Erisvaldo Vieira, Bruno |
| contents | We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $Σ$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of the boundary $H^{\partial M}$ bounded from below. We also prove a local rigidity result in the case $Σ$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $Σ$ and is isometric to $(-\varepsilon,\varepsilon)\times Σ, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partialΣ$ is totally geodesic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant Pessoa, Leandro F. Véras, Erisvaldo Vieira, Bruno Differential Geometry We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $Σ$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of the boundary $H^{\partial M}$ bounded from below. We also prove a local rigidity result in the case $Σ$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $Σ$ and is isometric to $(-\varepsilon,\varepsilon)\times Σ, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partialΣ$ is totally geodesic. |
| title | Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.10171 |