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arXiv · 2609.26357

Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity

Guoyi Xu
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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

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Guoyi Xu. 2026-09-22. Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity. https://arxiv.org/abs/2609.26357

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