Counting cusps on complete manifolds of finite volume
Peter Li, Jiaping Wang · Mathematical Research Letters · 2010
In this article, we consider complete, n-dimensional, Riemannian manifolds of finite volume.We assume that the Ricci curvature is bounded from below and normalized to have the lower bound given by Ric M ≥ -(n -1).Since M has finite volume, the constant functions are L 2 harmonic functions, implying that 0 is an eigenvalue for the L 2 -spectrum of the Laplacian.We define the quantity µ 1 (M ) by the Rayleigh quotientwhere the infimum is taken over all functions φ in the Sobolev space H 1 (M ) satisfying M φ = 0.This plays the role of a generalized first non-zero Neumann eigenvalue, although µ 1 (M ) might not necessarily be an eigenvalue.Note that