On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $\lambda_1<n$

Hang Chen; Peng Wang

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arXiv · 2405.10843

On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $\lambda_1<n$

Hang Chen·Peng Wang

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Abstract

In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $\lambda_1<n$ 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.

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BibTeX RIS

Hang Chen, Peng Wang. 2024-05-17. On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $\lambda_1<n$. https://arxiv.org/abs/2405.10843

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