On the analytic systole of Riemannian surfaces of finite type

TL;DR

This paper investigates the analytic systole of Riemannian surfaces of finite type, exploring inequalities involving the first Dirichlet eigenvalue and the systole, with implications for spectral geometry and curvature bounds.

Contribution

It analyzes the strictness of inequalities relating the analytic systole to the universal cover's spectrum and connects systole bounds with spectral properties under curvature constraints.

Findings

Brooks' inequality can be strict in certain cases.

Relation established between systole and the analytic systole under curvature bounds.

Improvement of previous results linking systole and spectral quantities.

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.