Spectral gap of random hyperbolic surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929282171797504 |
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| author | Anantharaman, Nalini Monk, Laura |
| author_facet | Anantharaman, Nalini Monk, Laura |
| contents | Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $λ_1=λ_1(X)$ bethe first non-zero eigenvalue of the Laplacian on $X$ or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~$α>0$,\begin{align*} \Pwp{λ_1 \leq \frac{1}{4} - α^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral gap of random hyperbolic surfaces Anantharaman, Nalini Monk, Laura Geometric Topology Spectral Theory Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $λ_1=λ_1(X)$ bethe first non-zero eigenvalue of the Laplacian on $X$ or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~$α>0$,\begin{align*} \Pwp{λ_1 \leq \frac{1}{4} - α^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers. |
| title | Spectral gap of random hyperbolic surfaces |
| topic | Geometric Topology Spectral Theory |
| url | https://arxiv.org/abs/2403.12576 |