Optimal constants in the diameter and volume-growth estimates

Determine the optimal constants in the residual-factor diameter estimate and in the volume-growth bounds for manifolds with nonnegative Ricci curvature and positive intermediate curvature, including the scalar-curvature case, except in the cases where the optimal constants are already known.

Background

The paper derives a universal diameter bound for the residual factor K in limits that split off an Euclidean factor under nonnegative Ricci curvature and a positive intermediate-curvature lower bound. It also proves universal volume-growth estimates of order Rm under the same curvature assumptions.

The remark identifies a quantitative refinement left unresolved: finding the best possible constants in both the residual-factor diameter estimate and the principal volume-growth theorems. The statement is explicitly limited to cases other than the few for which the optimal constants are known.

References

The optimal constant $C(n,m)$ in eqn:diamK is presently unknown. The optimal constant $C$ in Theorem \ref{thm:scalar-intro} and Theorem \ref{thm:main}, except in a few cases, is also unknown.

eqn:diamK:

$\diam K\leq C(n,m)\sigma^{-1}. $

— Universal Volume Growth Bounds from Positive Intermediate Curvature  (2608.14507 - Antonelli, 14 Aug 2026) in Unnumbered remark following the discussion of equation (\eqref{eqn:diamK}), Section 3, “Critical scales and effective rank improvement”

In the normalization used in this paper, Bray's conjecture is the following. For each $n\ge 3$, there exists a constant $\Lambda_n>1$, depending only on $n$, such that every closed $n$-dimensional Riemannian manifold $(N, g)$ satisfying $$ \operatorname{Ric}_g\ge (n-1)g, \qquad R_g\ge n(n-1)\Lambda_n, $$ also satisfies $$ \frac{|N|_g}{|\mathbb Sn|} \le \Lambda_n{-\frac n2}. $$

— An improved volume bound under Ricci and scalar curvature lower bounds  (2608.19196 - Kwong, 19 Aug 2026) in Section 1, Introduction