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.
"The Number of Cusps of Complete Riemannian Manifolds with Finite Volume." Taiwanese J. Math. 22 (6) 1403 - 1425, December, 2018. https://doi.org/10.11650/tjm/180604