Polar factorization of maps on Riemannian manifolds

By Robert J. McCann. 2001

563 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Equivariant Flows: exact likelihood generative learning for symmetric densities (2020, background): “Gradient flows (not necessarily G-equivariant) can map any ρ onto any ρ′ over Rn as long as both densities do not vanish (Benamou & Brenier, 2000; McCann, 2001) and have been discussed in the context of density estimation (Zhang et al., 2018; Papamakarios et al., 2019).”
Existence and stability results in the L 1 theory of optimal transportation (2003, background): “See also [ 34 ] for the extension of these results to a Riemannian setting.”
Monge's transport problem on a Riemannian manifold (2001, background): “Our starting point is slightly different, however, since we do not begin from a solution to the minimization problem (3) in which the integrand has been replaced by a strictly convex function d (p > 1) of the geodesic distance d(x, r(x)) [14].”
A User’s Guide to Optimal Transport (2013, background): “33 concerning optimal maps in Riemannian manifolds for the case costDsquared distance [64].”
Ricci curvature for metric-measure spaces via optimal transport (2004, methodology): “In the past few years, some regularity results for optimal transport on R have been extended to Riemannian manifolds [18, 29].”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, background): “A pioneering result in this direction is due to Cordero-Erausquin, McCann and Schmuckenschläger [18] who extended the Borell-Brascamp-Lieb inequalities to Riemannian manifolds by using the OMT-theory; in fact, this work stood at the basis of the papers of [42, 56, 57].”
Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions (2007, unknown): “After [7], McCann [46] used the generalized Legendre transform (5.”
On the regularity of solutions of optimal transportation problems (2009, background): “References concerning the existence of optimal map by Monge-Kantorovitch duality are [1] for the cost |x − y|(2), [16] and [8] for general costs, [23] for the Riemannian case, otherwise the book [32] offers a rather complete reference on the topic.”
Optimal transportation with infinitely many marginals (2012, unknown): “By McCann’s theorem [19], whenever ν ∈ Pac(M), there is a unique optimal map between ν and μ.”
OPTIMAL MAPS IN NON BRANCHING SPACES WITH RICCI CURVATURE BOUNDED FROM BELOW (2012, background): “Since then, such result has been generalized in many directions, the most important one being the one of McCann ([9]) proving the analogous result on Riemannian manifolds.”
Finsler interpolation inequalities (2009, methodology): “taneously to the Monge problem) in a Euclidean space is represented as the gradient of a convex function, and McCann [ 22 ] successfully extended it to Riemannian manifolds using c-concave functions.”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, methodology): “According to McCann [24], the optimal transport map between f(x) vol (dx) and g(y) vol (dy) takes the form”
Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna (2013, background): “From the results of [18], in the case where c = d/2, the optimal map can be expressed as a so-called gradient map, i.”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “Here differentiability of the cost function is crucial; for costs such as Riemannian distance squared, the desired uniqueness is known [29] [97], but the cost fails to be differentiable at the cut locus.”
Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media (2006, background): “In fact, (21) is nothing but superdifferentiability of the distance dist, which holds on Riemannian manifolds (see [42]).”