Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian

TL;DR

This paper establishes a connection between Bishop-Gromov inequalities and lower bounds of coarse Ricci curvature, providing estimates for Laplacian eigenvalues on metric measure spaces.

Contribution

It introduces a method to derive lower bounds of coarse Ricci curvature from Bishop-Gromov inequalities and links these bounds to eigenvalue estimates of the Laplacian.

Findings

Bishop-Gromov inequality implies a lower bound of coarse Ricci curvature.

Lower bounds of coarse Ricci curvature lead to eigenvalue estimates of the Laplacian.

The results apply to metric measure spaces with curvature bounds.

Abstract

We prove that a Bishop-Gromov inequality gives a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.