Integral of Scalar Curvature under Volume Collapse

We construct a smooth complete metric on ${\mathbb{R}}^3$ with strictly positive Ricci curvature and ${\displaystyle \underset{s\to \mathrm{\infty }}{\lim }{s}^{-1}{\int }_{B_p(s)}\text{Sc}\text{dd}\mu =\mathrm{\infty }}$. 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.