Shing-Tung Yau’s asymptotic scalar-curvature integral question and Naber’s local Shing-Tung Yau conjecture

Yau asked whether a natural scalar-curvature integral remains bounded on complete manifolds with nonnegative Ricci curvature; Naber proposed related noncollapsed and local formulations. The general question and Naber’s conjectures had remained unresolved.

Known results

• Nonnegative sectional curvature gives the asymptotic estimate; Xu obtained a three-dimensional bound for manifolds with a pole (2022 source).
• Conditional $O(r^{n-2})$ bounds follow from a $\mathbb{Z}^{n-2}$ subgroup in $\pi_1(M)$ and from local conformal flatness (June 2026).
• In dimension $3$, quadratic scalar-curvature decay yields the sharp bound $8\pi(1-\operatorname{AVR}(g))$ (June 2026).
• The locally conformally flat, nonnegative-Ricci case was claimed solved, but the unrestricted problem remained open (June 2026).

September 2026 counterexamples

A September 2026 preprint claims a smooth complete metric with infinite asymptotic scalar-curvature integral and compact collapsing examples, giving negative answers to Yau’s question and Naber’s local conjecture. The claims are unrefereed.

Current status (as of September 2026): A preprint claims both conjectures are false, but the counterexamples are unverified; the earlier general problem therefore has no independently confirmed resolution.