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

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Villani, Regularity of optimal transport and cut locus (DCDS 30, 2011) evidence for this event

Theorem 1.1: if $(M,g)$ is smooth, connected, compact, boundaryless, of dimension $n\ge2$, and satisfies weak MTW on orthogonal pairs, then every tangent injectivity domain $I(x)$ 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 $c$-convex potential and $C^{1,1}$ 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 $I(x)$, noncompact manifolds, or constants uniform over varying metrics. parent of this event

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

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Lean comparator statement: BiholderTransport.lean (companion) evidence for this event

VibeMathed record: Villani's conjecture: weak Ma-Trudinger-Wang curvature implies convex tangent injectivity domains on compact manifolds evidence for this event

Lean file: OAI/Geometry/WeakMTW/Convexity.lean evidence for this event

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

Companion: Uniform Bi-Holder Transport from Weak MTW evidence for this event

Lean comparator statement: WeakMTWGlobalSupport.lean evidence for this event