The Number of Cusps of Complete Riemannian Manifolds with Finite Volume

In this paper, we count the number of cusps of complete Riemannian manifolds M with finite volume. When M is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume V of M if some geometric conditions hold true. Moreover, we use the nonlinear theory of the p-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.