Integral of Scalar Curvature under Volume Collapse

Abstract

We construct a smooth complete metric on ${\mathbb{R}}^3$ with strictly positive Ricci curvature and $\displaystyle \lim_{s\rightarrow\infty}s^{-1}\int_{B_p(s)}\Sc\ddμ= \infty$. 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.