A conformal invariant and its application to the nonexistence of minimal submanifolds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914665631580160 |
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| author | Chen, Hang |
| author_facet | Chen, Hang |
| contents | Let $(M^m,g)$ be an $m$-dimensional closed Riemannian manifold with non-negative sectional curvatures, $m\ge 3$. We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on $m$, then there are no closed $n$-dimensional stable minimal submanifolds in $M$ for all $ξ(m)\le n\le m-2$, where $ξ(m)=1$ when $3\le m\le 5$ and $ξ(m)=2$ when $m\ge 6$. In particular, a conformal $m$-sphere with non-negative sectional curvatures does not admit any closed $n$-dimensional stable minimal submanifold for all $ξ(m)\le n\le m-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09724 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A conformal invariant and its application to the nonexistence of minimal submanifolds Chen, Hang Differential Geometry 53C40, 53C42, 53C18 Let $(M^m,g)$ be an $m$-dimensional closed Riemannian manifold with non-negative sectional curvatures, $m\ge 3$. We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on $m$, then there are no closed $n$-dimensional stable minimal submanifolds in $M$ for all $ξ(m)\le n\le m-2$, where $ξ(m)=1$ when $3\le m\le 5$ and $ξ(m)=2$ when $m\ge 6$. In particular, a conformal $m$-sphere with non-negative sectional curvatures does not admit any closed $n$-dimensional stable minimal submanifold for all $ξ(m)\le n\le m-2$. |
| title | A conformal invariant and its application to the nonexistence of minimal submanifolds |
| topic | Differential Geometry 53C40, 53C42, 53C18 |
| url | https://arxiv.org/abs/2310.09724 |