Estimates for low Steklov eigenvalues of surfaces with several boundary components
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909132820316160 |
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| author | Perrin, Hélène |
| author_facet | Perrin, Hélène |
| contents | In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $σ_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the geometry of $M$ away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01043 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Estimates for low Steklov eigenvalues of surfaces with several boundary components Perrin, Hélène Differential Geometry 35P15, 58C40 In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $σ_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the geometry of $M$ away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface. |
| title | Estimates for low Steklov eigenvalues of surfaces with several boundary components |
| topic | Differential Geometry 35P15, 58C40 |
| url | https://arxiv.org/abs/2211.01043 |