Improved relative volume comparison for integral Ricci curvature and applications to volume entropy

TL;DR

This paper develops improved volume comparison theorems for integral Ricci curvature, leading to better estimates of volume entropy and applications to the algebraic entropy of fundamental groups, extending rigidity results.

Contribution

It introduces enhanced Bishop-Gromov volume comparisons for integral Ricci curvature and applies them to improve volume entropy estimates and rigidity results.

Findings

Enhanced volume comparison theorems for integral Ricci curvature.

Improved volume entropy estimates based on these comparisons.

Extension of almost minimal volume rigidity to integral Ricci curvature.

Abstract

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of \cite{BBCG} to integral Ricci curvature.