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, Laplacian eigenvalue of geodesic balls and the total volume of a closed Riemannian manifold N , or manifold with some finiteness condition. For example, we prove that if the average scalar curvature is larger than the lower bound of the Ricci curvature, then we can improve the Bishop–Gromov estimate on the average volume of the metric balls of any size. We also prove a comparison theorem of the average total mean curvature of geodesic spheres of radius up to $$\textrm{inj}(N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>inj</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , without assuming any Ricci curvature bound. This shows that the average scalar curvature impacts the geometry at a macroscopic level, which is related to a question posed by Gromov about the possibility of integrating the infinitesimal volume comparison property of the scalar curvature to achieve comparisons for geodesic balls of all sizes.