Volume gap for minimal submanifolds in spheres
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915417848545280 |
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| author | Ge, Jianquan Li, Fagui |
| author_facet | Ge, Jianquan Li, Fagui |
| contents | For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+φ_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where $φ_{p}(x):=\langle f(x),p\rangle$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and ${\rm Vol}$ denotes the Riemannian volume functional, and each equality holds if and only if $M$ is totally geodesic. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04654 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Volume gap for minimal submanifolds in spheres Ge, Jianquan Li, Fagui Differential Geometry 53C42, 53C24 For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+φ_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where $φ_{p}(x):=\langle f(x),p\rangle$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and ${\rm Vol}$ denotes the Riemannian volume functional, and each equality holds if and only if $M$ is totally geodesic. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained. |
| title | Volume gap for minimal submanifolds in spheres |
| topic | Differential Geometry 53C42, 53C24 |
| url | https://arxiv.org/abs/2210.04654 |