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arXiv · 1611.00925

On the analytic systole of Riemannian surfaces of finite type

Werner Ballmann·Henrik Matthiesen·Sugata Mondal
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Abstract

In our previous work we introduced, for a Riemannian surface $S$, the quantity $ \Lambda(S):=\inf_F\lambda_0(F)$, where $\lambda_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $\Lambda(S)\ge\lambda_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $\Lambda(S)$ with the systole, improving a result by the last named author.

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Werner Ballmann, Henrik Matthiesen, Sugata Mondal. 2016-11-03. On the analytic systole of Riemannian surfaces of finite type. https://arxiv.org/abs/1611.00925

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