A new characterization for Clifford hypersurfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916145727012864 |
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| author | Cui, Qing Peñafiel, Carlos |
| author_facet | Cui, Qing Peñafiel, Carlos |
| contents | For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $σ_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $σ_k \ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $σ_2 \ge 16$. When $n\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\ge 2$, there exists a positive constant $δ_k(n)<n$, if $|A|^2\ge δ_k(n)$, we have $σ_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new characterization for Clifford hypersurfaces Cui, Qing Peñafiel, Carlos Differential Geometry For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $σ_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $σ_k \ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $σ_2 \ge 16$. When $n\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\ge 2$, there exists a positive constant $δ_k(n)<n$, if $|A|^2\ge δ_k(n)$, we have $σ_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface. |
| title | A new characterization for Clifford hypersurfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2403.01701 |