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Absolute continuity of Wasserstein geodesics in the Heisenberg group |
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| Authors: | A. Figalli N. Juillet |
| Affiliation: | a Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, CNRS UMR 6621 Parc Valrose, 06108 Nice Cedex 02, France b Institut Fourier BP 74, UMR 5582, Université Grenoble 1, 38402 Saint-Martin-d'Hères Cedex, France c Institut für Angewandte Mathematik, Universität Bonn, Poppelsdorfer Allee 82, 53115 Bonn, Germany |
| Abstract: | In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261-301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the end-points is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below. |
| Keywords: | Optimal transport Wasserstein geodesic Absolute continuity Heisenberg group Alexandrov spaces |
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