Effect of the average scalar curvature on Riemannian manifolds

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 $\mathrm{inj}(N)$.