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

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Abstract

We prove existence and uniqueness of optimal maps on \(\mathsf{RCD}^*(K,N)\) 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 \(\mathsf{RCD}^*(K,N)\) bounds.

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Acknowledgments

This paper was partly written during the program “Interactions Between Analysis and Geometry” at the Institute for Pure and Applied Mathematics (IPAM) at University of California, Los Angeles. The authors thank the institute for the excellent research environment. T.R. also acknowledges the support of the Academy of Finland Project No. 137528.

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Author notes

  1. Nicola Gigli

    Present address: SISSA, Trieste, Italy

Authors and Affiliations

  1. University of Nice, Nice, France

    Nicola Gigli

  2. University of Jyväskylä, Jyväskylä, Finland

    Tapio Rajala

  3. Universität Bonn, Bonn, Germany

    Karl-Theodor Sturm

Authors
  1. Nicola Gigli
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  2. Tapio Rajala
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  3. Karl-Theodor Sturm
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Correspondence to Nicola Gigli.

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Gigli, N., Rajala, T. & Sturm, KT. Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below. J Geom Anal 26, 2914–2929 (2016). https://doi.org/10.1007/s12220-015-9654-y

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  • DOI: https://doi.org/10.1007/s12220-015-9654-y

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