Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910160006414336 |
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| author | Jiao, Xiaoxiang Zou, Wenduo |
| author_facet | Jiao, Xiaoxiang Zou, Wenduo |
| contents | Given a closed Riemannian manifold $(M^{n+1},g)$,$3\leq n+1\leq7$.In this paper,we will prove that for any $c>0$,suppose the number of closed $c-CMC$ hypersurfaces is finite,then there exists a metric $h$ on $M$ such that the $c-CMC$ hypersurfaces in $(M,g)$ are also $c-CMC$ hypersurfaces in $(M,h)$ and the number of $c-CMC$ hypersurfaces in $(M,h)$ is strictly greater than the number of $c-CMC$ hypersurfaces in $(M,g)$.Moreover,we will give a precise upper bound for the $L^{\frac{n+1}{2}}$ norm of $(g-h)$,which depends on the metric $g$ and the number of $c-CMC$ hypersurfaces in $(M,g)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics Jiao, Xiaoxiang Zou, Wenduo Differential Geometry Given a closed Riemannian manifold $(M^{n+1},g)$,$3\leq n+1\leq7$.In this paper,we will prove that for any $c>0$,suppose the number of closed $c-CMC$ hypersurfaces is finite,then there exists a metric $h$ on $M$ such that the $c-CMC$ hypersurfaces in $(M,g)$ are also $c-CMC$ hypersurfaces in $(M,h)$ and the number of $c-CMC$ hypersurfaces in $(M,h)$ is strictly greater than the number of $c-CMC$ hypersurfaces in $(M,g)$.Moreover,we will give a precise upper bound for the $L^{\frac{n+1}{2}}$ norm of $(g-h)$,which depends on the metric $g$ and the number of $c-CMC$ hypersurfaces in $(M,g)$. |
| title | Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.19102 |