An improved volume bound under Ricci and scalar curvature lower bounds

Abstract

We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n,g)$ satisfies ${\mathrm{Ric}}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\epsilon )$, then its volume satisfies $$|N{|}_g\le \frac{1}{\sqrt{1+n\epsilon }}|{\mathbb{S}}^n|.$$ In fact, assuming only ${\mathrm{Ric}}_g\ge (n-1)g$, we can prove that $$\frac{|N{|}_g}{\left|{\mathbb{S}}^n\right|}\le \frac{1}{|N{|}_g}{\int }_N{\left(\frac{R_g}{n-1}-(n-1)\right)}^{-\frac{1}{2}}d{\mathrm{vol}}_g.$$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\epsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.

Corrected some typos and added an application to Q-curvature. 24 pages. Comments are welcome