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arXiv · 2609.10160

Integral of Scalar Curvature under Volume Collapse

Guoyi Xu
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Abstract

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 Shing-Tung Yau conjecture on compact manifolds. The noncompact example has zero asymptotic volume ratio, and the compact counterexamples have volumes tending to zero.

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Guoyi Xu. 2026-09-09. Integral of Scalar Curvature under Volume Collapse. https://arxiv.org/abs/2609.10160

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