For every integer n≥ 3, every complete n-dimensional Riemannian manifold (M^n,g) with operatornameRic_g≥ 0, and every point q∈ M, is the normalized scalar-curvature integral bounded as R→∞, namely, does limsup_R→∞R^2-nint_B_q(R)operatornameScal_g,dV_g<∞ hold?

Yau's question on normalized scalar curvature integrals

Yau posed the question in his 1990 problem list, asking whether normalized scalar-curvature integrals remain bounded on complete manifolds with nonnegative Ricci curvature.

Known results

Bo Yang’s 2013 examples answered the analogous question negatively for higher curvature quantities $k>1$ , while the scalar-curvature case $k=1$ remained open in the 2019 account.
In dimension three, Munteanu and Wang proved a positive formula under the additional assumption of a pole: the normalized integral has a finite limit for every center.
A 2026 preprint claims further finite bounds under topological or locally conformally flat hypotheses, but presents these as partial results.

September 2026 counterexample claim

An unrefereed preprint, Unbounded normalized scalar curvature integrals in dimension four, claims examples for which the normalized integral diverges from every center in all dimensions $n\geq4$ . If correct, this gives a negative answer to Yau’s question in those dimensions; the claim has not been independently verified in the retrieved sources.

Current status (as of September 2026): The scalar-curvature question is claimed to be settled negatively for $n\geq4$ , but that preprint is unverified; dimension three has only conditional positive results, and the unrestricted lower-dimensional picture remains open.

Yau's question on normalized scalar curvature integrals

Primary source

Additional references

Progress summary

Solutions 0