Continuity, curvature, and the general covariance of optimal transportation

By Young-Heon Kim, Robert J. McCann. 2007

182 citations recorded by the source corpus.

Let M and M be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ρ. For subdomains M and M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4(M × M) to guarantee smoothness of the optimal map pushing ρ forward to ρ; 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 pseudoRiemannian 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 principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and M of Rn. This maximum principle plays a key role in Loeper’s Holder 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/268864c50d856e8ceb4c355a5445973d5f10914f

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Citations in the corpus

Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “Necessity of Trudinger and Wang’s condition for continuity was shown by Loeper [42], who noted its covariance (as did [35] [49]) and some relations to curvature.”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Indeed the MTW condition can be proven to be coordinate invariant [73, 84, 87].”
Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna (2013, background): “the absence of boundary used in the saddle point argument. The case with boundary has been treated by Trudinger and Wang in [21]. An alternative argument to obtain ∂cφ= ∂φfrom Aw has been obtained in [15]. 5 Proof of Theorem 2.4 Strategy of the proof Most of the proof is contained in the following p”
Hölder regularity of optimal mappings in optimal transportation (2009, methodology): “Using the strict c-convexity in [20 ]( see also [ 15 ]), Loeper [17] obtained the C 1,α regularity for some”
Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2 (2009, background): “…[3, 4, 5, 9, 25, 26], in the case of the flat torus or nearly flat metrics [8, 10], on the standard sphere and its perturbations [11, 17, 19], and on manifolds with nonfocal cut locus [20] (see [28, Chapter 12] for an introduction to the problem of the regularity of the optimal tranport map for…”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, background): “Conversely, smoothness results have been obtained under various sets of assumptions including either the weak or the strong MTW condition [10, 16 , 19, 20, 22, 27]; such results are reviewed in [30, Chapter 12].”
Wasserstein Barycenters over Riemannian manifolds (2014, background): “In fact, as shown in [14], this property holds only on so-called nonnegatively cross curved spaces, which are spaces satisfying a certain fourth order condition on the metric; this fourth order condition is a strenthened variant of the celebrated Ma-Trudinger-Wang condition [27, 40] arising in the r”
C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two (2009, background): “…problem are measures supported on sets satisfying necessary smoothness and convexity conditions, a necessary and sufficient condition on the cost function for smoothness of the optimal map (for arbitrary smooth positive data) has been given in [7] [10] [5] (see also [4] or [12, Chapter 12]).”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “Thus −(vc, v) is a better candidate for a maximizer than −(u, v).”
Nearly Round Spheres Look Convex (2012, methodology): “Although the conclusion of Theorem 1.1 is natural and simple, our proof is quite indirect, since it is based on the Ma‐ Trudinger‐Wang curvature tensor (MTW tensor), introduced in [29] and further studied in [8, 14, 16, 22 , 23, 26, 28, 33, 34], in relation with the regularity theory of optimal tra”
STABILITY OF A 4TH-ORDER CURVATURE CONDITION ARISING IN OPTIMAL TRANSPORT THEORY (2008, methodology): “In [12, 14, 15, 17] strong versions of the MTW condition have been exploited to derive regularity estimates on the optimal transport map.”