Existence of multiple closed CMC hypersurfaces with small mean curvature
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914839428857856 |
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| author | Dey, Akashdeep |
| author_facet | Dey, Akashdeep |
| contents | Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC hypersurfaces with optimal regularity. More quantitatively, there exists a constant $γ_0$, depending on $g$, such that for all $c>0$, there exist at least $γ_0c^{-\frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$. This extends the theorem of Zhou and Zhu, where they proved the existence of at least one closed $c$-CMC hypersurface in $(M,g)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_00989 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Existence of multiple closed CMC hypersurfaces with small mean curvature Dey, Akashdeep Differential Geometry Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC hypersurfaces with optimal regularity. More quantitatively, there exists a constant $γ_0$, depending on $g$, such that for all $c>0$, there exist at least $γ_0c^{-\frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$. This extends the theorem of Zhou and Zhu, where they proved the existence of at least one closed $c$-CMC hypersurface in $(M,g)$. |
| title | Existence of multiple closed CMC hypersurfaces with small mean curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1910.00989 |