Gromov’s asymptotic volume growth conjecture

Determine whether every n-dimensional complete noncompact Riemannian manifold (M^n,g) with nonnegative Ricci curvature and scalar curvature satisfying Sc_g ≥ 1 obeys limsup_{r→∞} Vol(B(p,r))/r^{n−2} < ∞ for all points p ∈ M.

Background

The paper studies 3-dimensional complete noncompact manifolds with asymptotically nonnegative Ricci curvature and uniformly positive scalar curvature lower bounds, aiming to refine volume growth estimates. In this context, the authors recall a conjecture posed by Gromov concerning the asymptotic volume ratio for manifolds with nonnegative Ricci curvature and a uniform positive scalar curvature lower bound.

While this conjecture has been resolved in certain special cases—including a complete resolution in dimension three via work of Munteanu and Wang—the general n-dimensional case remains a central open problem. The present paper provides optimal asymptotic volume ratio bounds in three dimensions under broader hypotheses, thereby situating their results relative to Gromov’s conjecture.