Continuity, curvature, and the general covariance of optimal transportation

Abstract

Abstract. Let M andM be n-dimensional manifolds equipped with suitable Borel probability measures ρ andρ. For subdomains M andM of R n , Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C 4 (M ×M) to guarantee smoothness of the optimal map pushing ρ forward toρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M ×M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry.Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang's conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principle characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and M of R n . This maximum principle plays a key role in Loeper's Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.

A note on global regularity in optimal transportion

“…We need to apply these in c-sections of u whose diameters will be controlled by the strict convexity results in [4] and [24]. First we note that the ellipticity condition (1.2) and the c-convexity of the domain implies that u is c-convex in [21], (see also [20] and [8]), that is for any x 0 ∈ and y 0 = T u(x 0 ), we have…”

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

“…Thus the so called "double mountain above sliding mountain" property established in [26] is not true in our case and it is interesting to know whether the local c-subdifferential is still the same as the global c-subdifferential which is especially important in localizing our problem to understand the smoothness of u. As an attempt at answering this nontrivial question, we explore the problem of strict convexity in depth in order to obtain more refined information of the possible degeneracies and to describe the geometry of optimal transportation.…”

The Monge–Ampère equation and its link to optimal transportation

“…Apart from having its own interest, this general theory of Monge-Ampère type equations satisfying the MTW condition turns out to be useful also in the classical case. Indeed the MTW condition can be proven to be coordinate invariant [90,87,76]. This implies that, if u solves an equation of the form (1.2) with A satisfying the MTW condition (see (4.4)) and Φ is a smooth diffeomorphism, then u•Φ satisfies an equation of the same form with a new matrixà which still satisfies the MTW condition.…”

“…Some examples of manifolds satisfying the MTW condition have been found in [87,88,76,77,56,58]: -R n and T n satisfy MTW(0). -S n , its quotients (like RP n ), and its submersions (like CP n or HP n ), satisfy MTW(1).…”