A preprint in geometry reports a proof of Bray’s conjecture, which links lower bounds on curvature to an upper limit on volume. The manuscript says that, under the conjecture’s assumptions, a closed Riemannian manifold cannot have more volume than the standard n-sphere. The result appears in an arXiv math.DG preprint, version 1, dated 20 August 2026.
The work addresses no empirical sample. It is a universal mathematical claim about connected, closed, smooth Riemannian manifolds—geometric spaces described by a metric—with dimension n≥3. The question is whether a sufficiently small relaxation of the Ricci-curvature lower bound, alongside the stated scalar-curvature lower bound, still forces the same volume ceiling.
The volume ceiling at the center
Bray’s target is expressed through a positive, dimension-dependent number ε_n that is less than 1. It asks whether Ricci curvature at least ε_n(n−1)g and scalar curvature at least n(n−1) are enough to imply V_g(Mⁿ)≤|Sⁿ|. In ordinary language, the question is whether a manifold can satisfy those curvature lower bounds and still have a volume larger than the standard sphere.
The main theorem says the conjecture holds, but its final formulation uses a threshold ε̃_n rather than a single numerical value. For 0<ε≤ε̃_n, under the scaled curvature assumptions, it gives V_g(Mⁿ)≤(1+ε)^(-n/2)|Sⁿ|. The supplied analysis says this threshold is guaranteed to be positive, while no explicit general numerical value is reported.
This is a statement about the theorem’s permitted range, not a claim that every positive ε has the same consequence. The conclusion is tied to the closed-manifold setting, the dimension restriction and the two curvature hypotheses stated in the conjecture.
A proof built around entropy
To build the proof, the manuscript combines Perelman’s entropy with a spherical Pólya–Szegő rearrangement, Beckner’s inequality and a result by Ma and Wang. The strategy uses a failure of the volume inequality to derive an entropy comparison in which the manifold’s value is above the sphere’s.
A key estimate is stated for every positive scale τ. Under the curvature assumptions, the proof’s μ quantity is bounded below by the sphere’s corresponding μ value, plus a logarithm of the volume ratio and an ε-dependent curvature term. That translates the volume question into a comparison of the proof’s entropy quantities.
The spherical rearrangement supplies a test function with no larger Dirichlet energy, the quantity used here to measure how a function varies. The spherical entropy can then be minimized explicitly in the stated small-ε range. For 0<ε≤ε_ent(n), the manuscript gives a unique minimizing scale, τ_ε=1/[2(n−1)(1+ε)], and an infimum of ν(g_{Sⁿ})+(n/2)log(1+ε).
That calculation supplies the contradiction’s first half. Under the corollary’s assumptions, a volume greater than |Sⁿ|(1+ε)^(-n/2) implies ν(g)>ν(g_{Sⁿ}). This is a conditional mathematical implication: it describes what follows from a violation of the scaled bound, not an estimate drawn from observations.
The flow supplies the other side
The second half comes from volume-normalized Ricci flow, the evolution used in the manuscript to compare metrics over time. Its stated input is that this flow exists for all forward time and converges exponentially to a round metric of constant sectional curvature one.
If the metric has the sphere’s volume, and the normalized flow exists globally and converges smoothly to the stated round geometry, the paper gives ν(g)≤ν(g_{Sⁿ}). Thus the flow supplies the inequality in the opposite direction from the one produced by a volume violation.
The proof applies this flow comparison after rescaling the metric. By scale invariance, its stated argument carries the flow conclusion back to the original setting, giving ν(g)≤ν(g_{Sⁿ}) and contradicting the entropy lower bound associated with a volume excess.
A theorem with a defined perimeter
The result is therefore a theorem about a tightly defined mathematical setting: connected, closed, smooth Riemannian manifolds of dimension n≥3 subject to the stated Ricci and scalar-curvature lower bounds. Its conclusion is the conditional volume upper bound described in the conjecture.
The available result has clear boundaries. The positive threshold ε̃_n is existence-based rather than reported as an explicit general number, and the theorem is stated only for ε in its permitted range, not for arbitrary positive ε. The argument also depends on the cited rearrangement, inequality and flow inputs listed in the manuscript.
Those qualifications define how the paper should be read: as a conditional proof of a volume comparison under stated assumptions. The preprint also discloses AI-assisted tool use, principally ChatGPT, and says the theorem statements and proofs were verified and that the authors take responsibility for the contents.
Paper data and sources
Original title: On the proof of Bray's conjecture
Authors: Xumin Jiang, Mingxiang Li, Zhehui Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text