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arXiv · 2111.08985

Systole and small eigenvalues of hyperbolic surfaces

Pierre Jammes
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Abstract

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$χ$, and let $λ_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $λ_{-χ}>\frac14$. In this article, we prove that if thesystole of $S$ is greater than 3,46, then $λ_{-χ-1}>\frac14$.This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.

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Pierre Jammes. 2021-11-17. Systole and small eigenvalues of hyperbolic surfaces. https://arxiv.org/abs/2111.08985

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