Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity

For any complete noncompact manifolds $(M^n,g)$ of nonnegative sectional curvature, with $3\le n\le 6$, we obtain ${\displaystyle \underset{r\to \mathrm{\infty }}{\lim }{r}^{2-n}{\int }_{B_p(r)}{\text{Sc}}_{g}d{V}_g\le 8\pi {\omega }_{n-2}(1-\text{AVR}(M^n,g))}$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n,g)$ with positive-dimensional souls. With a pole, ${\displaystyle \underset{r\to \mathrm{\infty }}{\lim }{r}^{2-n}{\int }_{B_p(r)}{\text{Sc}}_{g}d{V}_g\le 4\pi {\omega }_{n-2}(1-\text{AVR}(M^n,g))}$ follows from an independent distance-sphere proof. We also prove the sharp total scalar-integral bound for closed manifolds whose c