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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2012, Issue 664


Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

Young-Heon Kim / Robert J. McCann
Published Online: 2011-06-28 | DOI: https://doi.org/10.1515/CRELLE.2011.105


The variant A3w of Ma, Trudinger and Wang's condition for regularity of optimal transportation maps is implied by the non-negativity of a pseudo-Riemannian curvature—which we call cross-curvature—induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature non-negativity is preserved for products of two manifolds; (2) both A3w and cross-curvature non-negativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the n-dimensional round sphere satisfies cross-curvature non-negativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature non-negativity (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.

About the article

Received: 2009-02-02

Revised: 2009-11-19

Published Online: 2011-06-28

Published in Print: 2012-03-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2012, Issue 664, Pages 1–27, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2011.105.

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