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On one-dimensionality of metric measure spaces


Author: Timo Schultz
Journal: Proc. Amer. Math. Soc. 149 (2021), 383-396
MSC (2010): Primary 53C23
DOI: https://doi.org/10.1090/proc/15162
Published electronically: October 21, 2020
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Abstract: In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $ CD(K,N)$ -space or an essentially non-branching $ MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching $ MCP(K,N)$-spaces.


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Additional Information

Timo Schultz
Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35, FI-40014 University of Jyvaskyla
Email: timo.m.schultz@jyu.fi

DOI: https://doi.org/10.1090/proc/15162
Keywords: Optimal transport, Ricci curvature, metric measure spaces, Gromov--Hausdorff tangents
Received by editor(s): December 11, 2019
Received by editor(s) in revised form: March 18, 2020, April 16, 2020, and April 23, 2020
Published electronically: October 21, 2020
Additional Notes: The author acknowledges the support by the Academy of Finland, project #314789.
Communicated by: Guofang Wei
Article copyright: © Copyright 2020 American Mathematical Society