## On one-dimensionality of metric measure spaces

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- Proc. Amer. Math. Soc.
**149**(2021), 383-396 Request permission

## 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.## References

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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
- MR Author ID: 1284121
- ORCID: 0000-0002-1311-0394
- Email: timo.m.schultz@jyu.fi
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 383-396 - MSC (2010): Primary 53C23
- DOI: https://doi.org/10.1090/proc/15162
- MathSciNet review: 4172613