Abstract. Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature. 1.

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Abstract. A path in a Riemannian manifold can be approximated by a path meeting only finitely many times the cut locus of a given point. The proof of this property uses recent works of Itoh–Tanaka and Li–Nirenberg about the differential structure of the cut locus. We present applications in the regularity theory of optimal transport.

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We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger–Wang condition is stable under C^4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.

Abstract. We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampère equations. 1.

Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

continuity for optimal multivalued mappings ∗

This abstract sketches a geometric framework proposed in [1] and its consequences concerning the general regularity theory for optimal mappings developed by Ma, Trudinger, Wang and Loeper, following pioneering work on special cost functions by (at least) Caffarelli, Delanoë, Huang, Guan, Gutierrez, Oliker, Urbas, and X-J Wang. Due to space limitations we do not attempt to cite the literature or give much historical context, referring the reader instead to our paper, except that we note a different approach to some of our results was discovered independently at about the same time by Trudinger & Wang in arXiv:math/0702807. For simplicity our assumptions here are more restrictive than required in [1]. Let M and ¯ M be domains with compact closure cl M ⊂ M ′ and cl ¯ M ⊂ ¯ M ′ in smooth manifolds M ′ and ¯ M ′. Suppose M and ¯ M are equipped with Borel probability measures ρ and ¯ρ, and let s ∈ C4 (Ω′) be the surplus ( = − transportation cost) defined on the product space Ω ′ = M ′ × ¯ M ′. The optimal transportation problem of Kantorovich is then to find the measure γ ≥ 0 on M × ¯ M which achieves the supremum