Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity

Abstract

For any complete noncompact manifolds $(M^n,g)$ of nonnegative sectional curvature, with $3\le n\le 6$, 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