# The number of cusps of complete Riemannian manifolds with finite volume

## TL;DR

This paper establishes bounds on the number of cusps in complete Riemannian manifolds with finite volume, using volume decay estimates, comparison theorems, and nonlinear $p$-Laplacian theory.

## Contribution

It provides new upper bounds on cusp counts based on volume and geometric conditions, extending previous results to measure spaces and nonlinear analysis.

## Key findings

- Number of cusps is bounded by volume under certain conditions
- Volume decay estimates are crucial for cusp counting
- Upper bounds are derived using $p$-Laplacian theory

## Abstract

In this paper, we will 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.