A Counterexample to Yau's Conjectured Asymptotic Scalar-Curvature Integral Bound

Abstract

A complete one-ended three-manifold with strictly positive Ricci curvature is constructed such that $$\underset{R\to \infty }{\mathrm{limsup}}\frac{1}{R}{\int }_{B(p,R)}ScaldV=+\infty .$$ The construction combines an explicit toric lens carrying a large scalar-curvature integral with a three-dimensional angular pair of pants and an adaptive sequence of hybrid blocks.

A counterexample by ChatGPT 5.6 Pro