Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity

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