Year: 2025
Type: article
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 $$\textrm{inj}(N)$$ inj ( N ) , 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.
Author: Kwok-Kun Kwong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4409545244