Abstract
In (J Differ Geom 103(1):1–13, 2016) we introduced, for a Riemannian surface S, the quantity \({\Lambda(S):={\rm \inf}_F\lambda_0(F)}\), 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 (J Reine Angew Math 357:101–114, 1985) implies \({\Lambda(S)\geq\lambda_0(\tilde{S})}\), 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 the main result of (Enseign Math 60(2):1–23, 2014).
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Ballmann, W., Matthiesen, H. & Mondal, S. On the analytic systole of Riemannian surfaces of finite type. Geom. Funct. Anal. 27, 1070–1105 (2017). https://doi.org/10.1007/s00039-017-0422-y
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DOI: https://doi.org/10.1007/s00039-017-0422-y