On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $λ_1<n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914800557096960 |
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| author | Chen, Hang Wang, Peng |
| author_facet | Chen, Hang Wang, Peng |
| contents | In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $λ_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in $\mathbb{S}^3$: a minimal torus in $\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and $r$-minimal hypersurfaces in a sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $λ_1<n$ Chen, Hang Wang, Peng Differential Geometry 53A10, 53C42 In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $λ_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in $\mathbb{S}^3$: a minimal torus in $\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and $r$-minimal hypersurfaces in a sphere. |
| title | On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $λ_1<n$ |
| topic | Differential Geometry 53A10, 53C42 |
| url | https://arxiv.org/abs/2405.10843 |