On onedimensionality of metric measure spaces
Schultz, T. (2021). On onedimensionality of metric measure spaces. Proceedings of the American Mathematical Society, 149(1), 383396. https://doi.org/10.1090/proc/15162
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2021Copyright
© 2020 American Mathematical Society
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 nonunique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a onedimensional 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 nonbranching MCP(K, N)space with some open set isometric to an interval, then it is a onedimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the GromovHausdorff 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 nonbranching MCP(K, N)spaces
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The author acknowledges the support by the Academy of Finland, project #314789License
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