For every integer n≥ 3, there exists a constant varepsilon_n∈(0,1) such that, for every connected, closed, smooth Riemannian manifold (M^n,g), if operatornameRic_g≥ varepsilon_n(n-1)g and R_g≥ n(n-1), then operatornameVol_g(M)≤ operatornameVol(S^n,g_round), where (S^n,g_round) is the standard unit n-sphere.

Bray’s conjecture

Bray’s 1997 conjecture says that suitable lower bounds on Ricci and scalar curvature force a sharp upper bound on the volume of a closed manifold, with the round sphere as the extremal case. Bray settled the three-dimensional case; the general higher-dimensional statement had remained open.

Known results

$n=3$ : Bray proved the conjecture, with estimates $0.134<\varepsilon_3<0.135$ .
$n=3$ : Gursky and Viaclovsky proved $\varepsilon_3\leq\frac12$ .
Brendle proved rigidity at the threshold $\varepsilon_3=\frac12$ .
Yuan, Zhang, and Kwong obtained near-spherical or stronger-assumption partial results, not the full conjecture.

August 20, 2026 claimed proof

Xumin Jiang, Mingxiang Li, and Zhehui Wang’s preprint On the proof of Bray’s conjecture states a theorem matching the conjecture and says it confirms Bray’s 1997 conjecture. The argument uses Perelman’s entropy and several sharp functional inequalities, but no independent verification, referee report, or error analysis was found. The authors disclose that ChatGPT assisted their work and say they verified the contents.

Current status (as of August 2026): The three-dimensional case is settled, while the higher-dimensional conjecture is claimed in a preprint but remains unverified.

Bray’s conjecture

Primary source

Additional references

Progress summary

Solutions 0