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Villani's conjecture: weak Ma-Trudinger-Wang curvature implies convex tangent injectivity domains on compact manifolds

Let (M,g)(M,g) be a compact connected Riemannian manifold without boundary, c=d2/2c=d^2/2, and I(x)⊂TxMI(x)\subset T_xM the open tangent injectivity domain. The Ma-Trudinger-Wang tensor S(x,v)(ξ,η)\mathfrak S_{(x,v)}(\xi,\eta) is a fourth derivative of cc; weak MTW (A3w) requires S(x,v)(ξ,η)≥0\mathfrak S_{(x,v)}(\xi,\eta)\ge0 whenever ⟨ξ,η⟩=0\langle\xi,\eta\rangle=0. Weak MTW is necessary for continuity of optimal transport maps (Loeper), and regularity theory (Figalli-Rifford-Villani) needed convex injectivity domains as an extra hypothesis. Villani proposed that weak MTW should by itself force convexity of every I(x)I(x); Loeper-Villani proved convexity under strong MTW with nonfocality, and Figalli-Gallouet-Rifford proved the implication for nonfocal manifolds, where cut points come strictly before conjugate points. Does weak MTW imply that I(x)I(x) is convex for every x∈Mx\in M, with no nonfocality assumption?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Riemannian geometry and optimal transport regularity
Posed by
C. Villani, Regularity of optimal transport and cut locus: from nonsmooth analysis to geometry to smooth analysis, Discrete Contin. Dyn. Syst. 30 (2011), Section 4.2
Year posed
2011
Years open
15y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: if (M,g)(M,g) is smooth, connected, compact, boundaryless, of dimension n≥2n\ge2, and satisfies weak MTW on orthogonal pairs, then every tangent injectivity domain I(x)I(x) is convex, and so is the closed minimizing domain; conjugate cut points are allowed. Theorem 1.2 adds global supporting mountains at every subgradient of a cc-convex potential and C1,1C^{1,1} intermediate-time geometry, with no density hypothesis. Companion: uniform bi-Holder estimates for optimal maps and their inverses over all densities pinched between fixed positive bounds. Not shown: strict convexity or boundary regularity of I(x)I(x), noncompact manifolds, or constants uniform over varying metrics.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies Lean formalizations of the convexity theorem and of the bi-Holder transport companion, produced as part of the same release.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript against Villani's conjecture as the paper quotes it, and the Lean statement lean/ComparatorChallenges/WeakMTWGlobalSupport.lean (OAI.WeakMTWGlobalSupport.current_main_convexity), listed in formalization.yaml; not rebuilt here. The Lean statement covers the headline: on a compact connected smooth Riemannian manifold of dimension at least two, weak MTW (defined from the fourth derivative of half squared distance along exponential curves, on orthogonal pairs) implies linear convexity of every open tangent injectivity domain. The exponential map is defined in Lean through minimizing geodesics, a modelling choice a reader may wish to audit. The companion's uniform bi-Holder transport theorem is formalized separately (BiholderTransport).

Sources

Changelog1 change

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