Effect of the average scalar curvature on Riemannian manifolds

TL;DR

This paper explores how the average scalar curvature influences various geometric properties of closed Riemannian manifolds, providing new comparison theorems and improved volume estimates based on curvature bounds.

Contribution

It introduces novel results linking average scalar curvature to geometric quantities and establishes comparison theorems under curvature conditions.

Findings

Improved Bishop-Gromov volume estimates when average scalar curvature exceeds a certain bound.

Monotonicity of a geometric integral related to scalar curvature.

Comparison theorem for the average total mean curvature of geodesic spheres.

Abstract

We investigate the effect of the average scalar curvature on the conjugate radius, average area of the geodesic spheres, average volume of the metric balls and the total volume of a closed Riemannian manifold $N$ (or more generally $N$ with finite volume whose negative Ricci curvature integral on $SN$ is finite). For example, we prove that if the average scalar curvature is larger than the lower bound of the normalized Ricci curvature, then we can improve the Bishop-Gromov estimate on the average volume of the metric balls of any size. We also prove the monotone decreasing property of a certain geometric integral when the average scalar curvature has a lower bound. This leads to a comparison theorem of the average total mean curvature of geodesic spheres of radius up to $\mathrm{inj}(N)$.