Second Variation Formula for Eigenvalue Functionals on Surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911105527316480 |
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| author | Karpukhin, Mikhail |
| author_facet | Karpukhin, Mikhail |
| contents | Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional and minimal surfaces in the sphere. It was later extended by A. El Soufi and S. Ilias to cover k-th eigenvalues and critical points in a fixed conformal class, where the latter correspond to harmonic maps to the sphere. These results, however, only contain first order information and cannot be used to determine whether a given critical metric a local maximiser or not. In the present paper we write down the second variation formula for critical metrics and show that the flat metric on the non-rhombic torus can never be a conformal maximiser for the first eigenvalue. Analogous results are proved in the context of the Steklov eigenvalues and flat metrics on a cylinder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second Variation Formula for Eigenvalue Functionals on Surfaces Karpukhin, Mikhail Spectral Theory Differential Geometry Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional and minimal surfaces in the sphere. It was later extended by A. El Soufi and S. Ilias to cover k-th eigenvalues and critical points in a fixed conformal class, where the latter correspond to harmonic maps to the sphere. These results, however, only contain first order information and cannot be used to determine whether a given critical metric a local maximiser or not. In the present paper we write down the second variation formula for critical metrics and show that the flat metric on the non-rhombic torus can never be a conformal maximiser for the first eigenvalue. Analogous results are proved in the context of the Steklov eigenvalues and flat metrics on a cylinder. |
| title | Second Variation Formula for Eigenvalue Functionals on Surfaces |
| topic | Spectral Theory Differential Geometry |
| url | https://arxiv.org/abs/2508.10465 |