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 (Nn, g) satisfies Ricg≥ (n−1)g and the scalar curvature Rg≥ n(n−1)(1+ε), then its volume satisfies lvert Nrvertg ≤ 1/(√(1+nε))lvert𝕊nrvert. In fact, assuming only Ricg≥(n−1)g, we can prove that |N|g/|𝕊n| ≤ 1/|N|g ∫N(Rg/(n−1)−(n−1))−1/2 d volg. 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 ε 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.
Related
- Scalar curvature, volume, and Weil-Petersson translation length
- Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature
- Integral of Scalar Curvature under Volume Collapse
- A Counterexample to Yau's Conjectured Asymptotic Scalar-Curvature Integral Bound
- Macroscopic scalar curvature bounds on surfaces
BibTeX
@misc{kwong2026improved,
title = {{An improved volume bound under Ricci and scalar curvature lower bounds}},
author = {Kwok-Kun Kwong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.19196},
eprint = {2608.19196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19196},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An improved volume bound under Ricci and scalar curvature lower bounds
AU - Kwok-Kun Kwong
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.19196
UR - https://arxiv.org/abs/2608.19196
AB - 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 Ric_g≥ (n−1)g and the scalar curvature R_g≥ n(n−1)(1+ε), then its volume satisfies lvert Nrvert_g ≤ 1/(√(1+nε))lvert𝕊^nrvert. In fact, assuming only Ric_g≥(n−1)g, we can prove that |N|_g/|𝕊^n| ≤ 1/|N|_g ∫_N(R_g/(n−1)−(n−1))^−1/2 d 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 ε 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.
ER -