Yau’s scalar-curvature integral conjecture

Establish whether every complete noncompact Riemannian manifold of dimension n at least 3 with nonnegative Ricci curvature satisfies a finite asymptotic bound for the normalized scalar-curvature integral, namely whether limsup as r tends to infinity of r^{2-n} times the integral of scalar curvature over B_r(p) is finite.

Background

The paper relates its volume-growth theorem to a broader problem posed by Yau: controlling the scalar-curvature integral on large metric balls of complete noncompact manifolds with nonnegative Ricci curvature. If this conjectural integral estimate held under the additional assumption of scalar curvature bounded below by 1, it would immediately imply the paper’s volume-growth estimate.

The authors explicitly state that Yau’s conjecture remains unresolved in every dimension n at least 3, making it a qualifying open problem under the stated inclusion criteria.