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 asks whether curvature lower bounds force a sharp upper bound on total volume in every dimension $n \ge 3$ . Bray proved the three-dimensional case; the higher-dimensional statement was previously open.

Known results

$n=3$ : Bray proved the volume comparison theorem (1997).
$n=3$ : Gursky and Viaclovsky established the relevant threshold bound $\varepsilon_3 \le \frac12$ .
Equality rigidity at the threshold was proved by Brendle.
Zhang and later work obtained partial higher-dimensional results, but not the full conjecture.

August 2026 claimed proof

The preprint On the proof of Bray's conjecture claims a conditional form of the conjecture and states that its theorem confirms Bray’s 1997 conjecture in the manuscript’s formulation. This is a significant claimed resolution, but the preprint has not yet been independently verified, and its precise equivalence with Bray’s original formulation remains to be established.

Current status (as of August 2026): The three-dimensional case is settled, while a new preprint claims the higher-dimensional conjecture in its formulation; full resolution remains unverified pending confirmation of the formulation’s equivalence and proof.

Bray’s conjecture

Progress summary

Primary source

Additional references

Solutions 0