Abstract. We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather's minimal measures.Several observations have recently renewed the interest for the classical topic of optimal mass transportation, whose origin is attributed to Monge a few years before the French revolution. The framework is as follows. A space M is given, which in the present paper will be a compact manifold, as well as a continuous cost function c(x, y) : M × M → R. Given two probability measures µ 0 and µ 1 on M, the mappings : M → M which transport µ 0 into µ 1 and minimize the total cost M c(x, (x)) dµ 0 are studied. It turns out, and it was the core of the investigations of Monge, that these mappings have very remarkable geometric properties, at least at a formal level.Only much more recently was the question of the existence of optimal objects rigorously solved by Kantorovich in a famous paper of 1942. Here we speak of optimal objects, and not of optimal mappings, because the question of existence of an optimal mapping is ill-posed, so that the notion of optimal objects has to be relaxed, in a way that nowadays seems very natural, and that was discovered by Kantorovich.Our purpose here is to continue the work initiated by Monge, recently awakened by Brenier and enriched by other authors, on the study of geometric properties of optimal objects. The cost functions we consider are natural generalizations of the cost c(x, y) = d(x, y) 2 considered by Brenier and many other authors. The book [35] gives some ideas on the applications expected from this kind of questions. More precisely, we consider a Lagrangian function L(x, v, t) : T M ×R → R which is convex in v and satisfies standard
L(γ (t),γ (t), t) dtwhere the minimum is taken over the set of curves γ : [0, 1] → M satisfying γ (0) = x and γ (1) = y. Note that this class of costs does not contain the very natural cost c(x, y) = d(x, y). Such costs are studied in another paper [8].Our main result is that the optimal transports can be interpolated by measured Lipschitz laminations, or geometric currents in the sense of Ruelle and Sullivan. Interpolations of transport have already been considered by Benamou, Brenier and McCann for less general cost functions, and with different purposes. Our methods are inspired by the theory of Mather, Mañé and Fathi on Lagrangian dynamics, and we will detail rigorously the relations between these theories. Roughly, they are exactly similar except that mass transportation is a Dirichlet boundary value problem, while Mather theory is a periodic boundary value problem. We will also prove, extending work of Brenier, Gangbo, McCann, Carlier, and others, that the optimal...
