Year: 2026
Type: preprint
Abstract: For any complete noncompact manifolds $(M^n, g)$ of nonnegative sectional curvature, with $3\le n\le6$, we obtain $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le8πω_{n-2}(1-\AVR(M^n,g))$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n, g)$ with positive-dimensional souls. With a pole, $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le4πω_{n-2}(1-\AVR(M^n,g))$ follows from an independent distance-sphere proof. We also prove the sharp total scalar-integral bound for closed manifolds whose c
Source: arXiv (Cornell University)
Author: Xu, Guoyi
Institution: Tsinghua University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G1678429644
OpenAlex ID: W7214119807