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

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.

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.