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
Publication date: 21/10/2020
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
Contributing organizations
Related projects
- Geometric Aspects of Sobolev Space Theory
- Rajala, Tapio
- Academy of Finland
Ministry reporting: Yes
Reporting Year: 2021
Preliminary JUFO rating: 2