Improved Relative Volume Comparison for Integral Ricci Curvature and Applications to Volume Entropy

Abstract

We give several Bishop–Gromov relative volume comparisons with integral Ricci curvature which improve the results in Petersen and Wei (Geom Funct Anal 7:1031–1045, 1997). 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 Aubry (Int Math Res Notes 10:1933–1946, 2009) and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of Bessières et al. (Duke Math J 161(1):29–67, 2012) to integral Ricci curvature.