Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916994114125824 |
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| author | Audet-Beaumont, Samuel |
| author_facet | Audet-Beaumont, Samuel |
| contents | We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces $S_p$ with a specific subgroup of isometry $G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$ for each prime $p$. We do so by gluing surfaces with boundary following the structure of the Cayley graph of $G_p$. We then exploit the properties of $G_p$ and $S_p$ in order to show that an irreducible representation of high degree (depending on $p$) acts on the eigenspace of functions associated with $σ_1(S_p)$, leading to the desired result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity Audet-Beaumont, Samuel Spectral Theory Differential Geometry 58J50, 35P15 We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces $S_p$ with a specific subgroup of isometry $G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$ for each prime $p$. We do so by gluing surfaces with boundary following the structure of the Cayley graph of $G_p$. We then exploit the properties of $G_p$ and $S_p$ in order to show that an irreducible representation of high degree (depending on $p$) acts on the eigenspace of functions associated with $σ_1(S_p)$, leading to the desired result. |
| title | Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity |
| topic | Spectral Theory Differential Geometry 58J50, 35P15 |
| url | https://arxiv.org/abs/2412.07692 |