On the proof of Bray's conjecture

Xumin Jiang, Mingxiang Li, Zhehui Wang

arXiv:2608.20215v1 Primary: math.DG

Abstract

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

Disclosure

“s inequality, we show that if (1.6) fails, then ν(g) is strictly greater than the entropy limit of the n-sphere. However, by applying Ma–Wang’s result [9], we prove that this cannot happen. Disclosure on AI assistance The authors used AI-assisted tools, principally ChatGPT. The authors verified all theorem statements, proofs, and take full responsibility for the contents of the paper. 2. Preliminaries In this section, we introduce four ke”
PDF page 2

Classification: Mixed or intermediate disclosed use
Multiplier: 5
Verified: 2026-08-22T03:16:43.727629Z

Structural counts

Pages 9 pdf

Theorems 5 source

Lemmas 2 source

Propositions 3 source

Corollaries 2 source

Definitions 0 source

Displayed equations 80 source

Bibliography entries 16 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file On_the_proof_of_Bray_s_conjecture_1.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.