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Guo-Yi Xu

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Preprint Sep 2026

Integral of Scalar Curvature under Volume Collapse

We construct a smooth complete metric on $\R^3$ with strictly positive Ricci curvature and $\displaystyle \lim_{s\rightarrow\infty}s^{-1}\int_{B_p(s)}\Sc\dd\mu= \infty$. This gives a negative answer to Shing-Tung Yau's question on the asymptotic scalar-curvature integral. We also give counterexamples to Naber's local S...

Guo-Yi Xu · 4 citations

Preprint Sep 2026

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 $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le8\pi\omega_{n-2}(1-\AVR(M^n,g))$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n, g)$ with po...

Guo-Yi Xu · 2 citations · ⚡2