Small eigenvalues of hyperbolic surfaces with many cusps

Will Hide; Joe Thomas

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arXiv · 2410.06093

Small eigenvalues of hyperbolic surfaces with many cusps

Will Hide·Joe Thomas

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Abstract

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show that for any $a>0$, there exists $b>0$ such that if $g<an$ and $X$ has genus $g$ and $n$ cusps, then $X$ has at least $b(2g+n-2)$ small eigenvalues. This saturates the sharp upper bound $2g+n-2$ of Otal and Rosas up to a constant multiplicative factor.

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Will Hide, Joe Thomas. 2024-10-08. Small eigenvalues of hyperbolic surfaces with many cusps. https://arxiv.org/abs/2410.06093

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