Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below

We prove existence and uniqueness of optimal maps on RCD ∗ ( K , N ) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD ∗ ( K , N ) bounds.

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Veröffentlicht in:	The Journal of Geometric Analysis 2016-10, Vol.26 (4), p.2914-2929
Hauptverfasser:	Gigli, Nicola, Rajala, Tapio, Sturm, Karl-Theodor
Format:	Artikel
Sprache:	eng
Schlagworte:	Abstract Harmonic Analysis Convex and Discrete Geometry Curvature Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics Uniqueness
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Beschreibung
Zusammenfassung:	We prove existence and uniqueness of optimal maps on RCD ∗ ( K , N ) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD ∗ ( K , N ) bounds.
ISSN:	1050-6926 1559-002X
DOI:	10.1007/s12220-015-9654-y