NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS

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NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS A. FIGALLI, L. RIFFORD, AND C. VILLANI Abstract. In this paper we continue the investigation of the regularity of op- timal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma–Trudinger–Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal trans- port map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity and review existing results. 1. Introduction Throughout this paper,M will stand for a smooth compact connected n-dimensional Riemannian manifold (n ≥ 2) with its metric tensor g, its geodesic distance d and its volume vol . Reminders and basic notation from Riemannian geometry (exponential map, cut locus, injectivity domain, etc.) are gathered in Appendix A. Let µ, ? be two probability measures onM and let c(x, y) = d(x, y)2/2. The Monge problem with measures µ, ? and cost c consists in finding a map T : M ?M which minimizes the cost functional ∫ M c(x, T (x)) dµ(x) under the constraint T_µ = ? (? is the image measure of µ by T ).

• cut locus


• riemannian manifold


• general cost functions


• trans- port map


• ma–trudinger–wang condition


• transport continuity


• since property


• implies ??


• compact connected

Sujets

• Cut locus
• Riemannian manifold

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 16
• Langue: English

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NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS

A. FIGALLI, L. RIFFORD, AND C. VILLANI

Abstract.this paper we continue the investigation of the regularity of op-In timal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma–Trudinger–Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal trans- port map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity and review existing results.

1.noitcudotrIn

Throughout this paper,Mwill stand for a smooth compact connectednid-snemanoil Riemannian manifold (n≥2) with its metric tensorg, its geodesic distancedand its volume vol . Reminders and basic notation from Riemannian geometry (exponential map, cut locus, injectivity domain, etc.) are gathered in Appendix A. Let νbe two probability measures onMand letc(x y) =d(x y) 2  Monge2. The problem with measures νand costcconsists in finding a mapT:M→Mwhich minimizes the cost functional R M c(x T(x))d(x) under the constraintT # =ν(ν is the image measure ofbyT). Ifdoes not give mass to countably (n−1)-rectifiable sets, then according to McCann [25] this minimizing problem has a solutionT, unique up to modification on a-negligible set; moreoverTtakes the formT(x) = exp x (∇ x ψ), whereψ:M→R is ac-convex function [27, Chapter 5]. In the sequel,will be absolutely continuous with respect to the volume measure, and its density will be bounded from above and below; so McCann’s theorem applies andTis uniquely determined up to modification on a set of zero volume. The regularity ofTis in general a more subtle problem which has received much attention in recent years [27, Chapter 12]. A first question is whether the optimal transport map can be expected to be continuous.

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A. FIGALLI, L. RIFFORD, AND C. VILLANI

Definition 1.1.We say thatMsatisfies thetransport continuity propertyeriv-a(bb ated (TCP)) 1 if, wheneverandνare absolutely continuous measures with respect to the volume measure, with densities bounded away from zero and infinity, the opti- mal transport mapTwith measures νand costcis continuous, up to modification on a set of zero volume.

The aim of this paper is to give necessary and sufficient conditions for (TCP). This problem involves two geometric conditions: the condition ofconvexity of in- jectivity domains, and theMa–Trudinger–Wang condition. These conditions were first introduced and studied by Ma, Trudinger and Wang [24] and Loeper [22] outside the Riemannian world; the natural Riemannian adaptation of these concepts was made precise in Loeper and Villani [23] and further developed by Figalli and Rifford [12]. Both conditions come in the form of more or less stringent variants. •We say thatMsatisfies(CI)(resp.(CI + )) if for anyx∈Mthe injectivity domain I(x)⊂T x M convex).is convex (resp. strictly •For anyx∈M,v∈I(x) and (ξ η)∈T x M×T x M, lety= exp x v, we define the Ma–Trudinger–Wang tensor at (x y), evaluated on (ξ η), by the formula S (xy) (ξ η) =− 2 3d sd 2 2  s=0 d td 2 2  t=0 c  exp x (tξ)exp x (v+sη)   Then we say thatMsatisfies(MTW)if

(1.1)∀(x v)∈T Mwithv∈I(x)∀(ξ η)∈T x M×T x M h hξ ηi x = 0 =⇒S (xy) (ξ η)≥0 i  If the last inequality in (1.1) is strict unlessξ= 0 orη= 0, thenMis said to satisfy thestrictMa–Trudinger–Wang condition(MTW + ). In the case of nonfocal Riemannian manifolds, that is, when the injectivity do- mains do not intersect the focal tangent cut locus, Loeper and Villani [23] showed that(MTW + )implies(CI + ), (TCP), and then further regularity properties. The much more tricky focal case was first attacked in [12] and pursued in a series of works by the authors [13, 14, 15]. In presence of focalization, the robustness of the results becomes an issue, since then the stability of conditions(MTW + )and (CI + ) In [13] it was shown thatis not guaranteed under perturbations of the metric. perturbations of the sphere satisfy(MTW + )and(CI + ) [14] we showed that. In in dimensionn= 2 these two conditions are stable under perturbation, provided

1 This definition differs slightly from that in [12, Definition 1.1].