We show that the Laplacian of a Riemannian metric on a closed surface S with Euler characteristic χ(S) < 0 has at most −χ(S) small eigenvalues.

Abstract. Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and Euler characteristic χ(S) < 0 has at most −χ(S) small eigenvalues.

No abstract

Abstract. In (J Differ Geom 103(1):1-13, 2016) we introduced, for a Riemannian surface S, the quantity Λ(S) := inf F λ 0 (F ), where λ 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 Λ(S) ≥ λ 0 (S), the bottom of the spectrum of the universal coverS. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate Λ(S) with the systole, improving the main result of (Enseign Math 60(2):1-23, 2014).