On the analytic systole of Riemannian surfaces of finite type

Abstract

In (J Differ Geom 103(1):1–13, 2016) we introduced, for a Riemannian surface S, the quantity {Lambda(S):={rm inf}_Flambda_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 (J Reine Angew Math 357:101–114, 1985) implies {Lambda(S)geqlambda_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 the main result of (Enseign Math 60(2):1–23, 2014).