A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914413479460864 |
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| author | Gloria-Picazzo, Ricardo Wu, Yingying Yau, Shing-Tung |
| author_facet | Gloria-Picazzo, Ricardo Wu, Yingying Yau, Shing-Tung |
| contents | We explain how the spectrum of a closed embedded surface $Σ\subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $Ω\subset \mathbb{R}^3$ with $\partial Ω= Σ$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $Σ$, such that $λ_k^D(Ω)^{3/2}/(λ_k(Σ)\sqrt{λ_1(Σ)}) \ge K_g$, where $λ_k(Σ)$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $Σ$ and $λ_k^D(Ω)$ denotes the $k$-th eigenvalue of the Laplacian on $Ω$ with Dirichlet boundary conditions. Moreover, we explicitly obtain the dependence of $K_g$ on the genus, showing that $K_g \propto (g+1)^{-1}$, and we determine the optimal constant $K_0$ for $k=1$ in the genus-zero case. A generalized version of this result in arbitrary dimension is also provided for domains whose boundaries have nonnegative Ricci curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces Gloria-Picazzo, Ricardo Wu, Yingying Yau, Shing-Tung Differential Geometry Spectral Theory 58J50 (Primary), 35P15, 53C21 (Secondary) We explain how the spectrum of a closed embedded surface $Σ\subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $Ω\subset \mathbb{R}^3$ with $\partial Ω= Σ$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $Σ$, such that $λ_k^D(Ω)^{3/2}/(λ_k(Σ)\sqrt{λ_1(Σ)}) \ge K_g$, where $λ_k(Σ)$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $Σ$ and $λ_k^D(Ω)$ denotes the $k$-th eigenvalue of the Laplacian on $Ω$ with Dirichlet boundary conditions. Moreover, we explicitly obtain the dependence of $K_g$ on the genus, showing that $K_g \propto (g+1)^{-1}$, and we determine the optimal constant $K_0$ for $k=1$ in the genus-zero case. A generalized version of this result in arbitrary dimension is also provided for domains whose boundaries have nonnegative Ricci curvature. |
| title | A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces |
| topic | Differential Geometry Spectral Theory 58J50 (Primary), 35P15, 53C21 (Secondary) |
| url | https://arxiv.org/abs/2603.21035 |