A1 Journal article (refereed)
On one-dimensionality of metric measure spaces (2021)

Schultz, T. (2021). On one-dimensionality of metric measure spaces. Proceedings of the American Mathematical Society, 149(1), 383-396. https://doi.org/10.1090/proc/15162

JYU authors or editors

Publication details

All authors or editors: Schultz, Timo

Journal or series: Proceedings of the American Mathematical Society

ISSN: 0002-9939

eISSN: 1088-6826

Publication year: 2021

Volume: 149

Issue number: 1

Pages range: 383-396

Publisher: American Mathematical Society (AMS)

Publication country: United States

Publication language: English

DOI: https://doi.org/10.1090/proc/15162

Publication open access: Not open

Publication channel open access:

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/73037

Publication is parallel published: https://arxiv.org/abs/1912.01579

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

Keywords: differential geometry; measure theory; metric spaces

Free keywords: optimal transport; Ricci curvature; metric measure spaces; Gromov--Hausdorff tangents

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Ministry reporting: Yes

Reporting Year: 2021

Preliminary JUFO rating: 2