On the analytic systole of Riemannian surfaces of finite type

In our previous work we introduced, for a Riemannian surface $S$, the quantity $\mathrm{\Lambda }(S):={\inf }_F{\lambda }_0(F)$, 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 $\mathrm{\Lambda }(S)\ge {\lambda }_0(\tilde{S})$, 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 $\mathrm{\Lambda }(S)$ with the systole, improving a result by the last named author.