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30
Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres
"... Abstract. The variant A3w of Ma, Trudinger and Wang’s condition for regularity of optimal transportation maps is implied by the nonnegativity of a pseudoRiemannian curvature — which we call crosscurvature — induced by the transportation cost. For the Riemannian distance squared cost, it is shown t ..."
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Cited by 17 (6 self)
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Abstract. The variant A3w of Ma, Trudinger and Wang’s condition for regularity of optimal transportation maps is implied by the nonnegativity of a pseudoRiemannian curvature — which we call crosscurvature — induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) crosscurvature nonnegativity is preserved for products of two manifolds; (2) both A3w and crosscurvature nonnegativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the ndimensional round sphere satisfies crosscurvature nonnegativity. From these results, a large new class of Riemannian manifolds satisfying crosscurvature nonnegativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for certain Riemannian submersions of the sphere such as the complex projective spaces CPn. 1.
Nearly round spheres look convex
 of Progress in Mathematics
"... Abstract. We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains — so M appears uniformly convex in any exponential chart. The proof is based on the Ma–Trudinger–Wang nonlocal curvature tensor, which originates from t ..."
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Cited by 13 (5 self)
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Abstract. We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains — so M appears uniformly convex in any exponential chart. The proof is based on the Ma–Trudinger–Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds
 Int. Math. Res. Not. IMRN 2008, Art. ID rnn120
"... 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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Cited by 13 (3 self)
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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.
An approximation lemma about the cut locus, with applications in optimal transport theory
 Methods Appl. Anal
"... 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 regu ..."
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Cited by 10 (4 self)
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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.
On the MaTrudingerWang Curvature on Surfaces
"... 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 positi ..."
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Cited by 8 (2 self)
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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.
PseudoRiemannian geometry calibrates optimal transportation
 Math. Research Lett
"... Abstract. Given a transportation cost c: M × ¯ M → R, optimal maps minimize the total cost of moving masses from M to ¯ M. We find, explicitly, a pseudometric and a calibration form on M × ¯ M such that the graph of an optimal map is a calibrated maximal submanifold, and hence has zero mean curva ..."
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Cited by 6 (5 self)
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Abstract. Given a transportation cost c: M × ¯ M → R, optimal maps minimize the total cost of moving masses from M to ¯ M. We find, explicitly, a pseudometric and a calibration form on M × ¯ M such that the graph of an optimal map is a calibrated maximal submanifold, and hence has zero mean curvature. We define the mass of spacelike currents in spaces with indefinite metrics. 1.
FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS
"... Abstract. In this series of lectures we introduce the MongeKantorovich 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, ..."
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Cited by 5 (2 self)
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Abstract. In this series of lectures we introduce the MongeKantorovich 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 MongeAmpère 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.