Weak MTW curvature gives convexity and regular optimal transport

On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous. (Lean)

Manuscripts (2)

Global Support and Convex Injectivity Domains under Weak MTW

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026}

We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.

Uniform Bi-Holder Transport from Weak MTW

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026}

We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.

Result 360 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.