A lower bound for the first eigenvalue of a minimal hypersurface in the sphere
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909213639311360 |
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| author | Jiménez, Asun Chinchay, Carlos Tapia Zhou, Detang |
| author_facet | Jiménez, Asun Chinchay, Carlos Tapia Zhou, Detang |
| contents | Let $Σ$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $Λ=\max\limits_Σ|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $Σ$ satisfies
$$λ_1(Σ)> \dfrac{m}{2}+\frac{m(m+1)}{32(12Λ+m+11)^2+8},$$ and $λ_1(Σ)=m$, when $Λ\le\sqrt{m}$.
In particular, this estimate improves the one obtained recently in \cite{duncan2023improved}. The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of $Σ$ in the spirit of \cite{choi1983first}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound for the first eigenvalue of a minimal hypersurface in the sphere Jiménez, Asun Chinchay, Carlos Tapia Zhou, Detang Differential Geometry 53C42, 53C44 Let $Σ$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $Λ=\max\limits_Σ|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $Σ$ satisfies $$λ_1(Σ)> \dfrac{m}{2}+\frac{m(m+1)}{32(12Λ+m+11)^2+8},$$ and $λ_1(Σ)=m$, when $Λ\le\sqrt{m}$. In particular, this estimate improves the one obtained recently in \cite{duncan2023improved}. The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of $Σ$ in the spirit of \cite{choi1983first}. |
| title | A lower bound for the first eigenvalue of a minimal hypersurface in the sphere |
| topic | Differential Geometry 53C42, 53C44 |
| url | https://arxiv.org/abs/2405.20545 |