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
MathSciNet review:
4172613
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- Guillaume Carlier, Luigi De Pascale, and Filippo Santambrogio, A strategy for non-strictly convex transport costs and the example of $\|X-Y\|^P$ in $\Bbb R^2$, Commun. Math. Sci. 8 (2010), no. 4, 931–941. MR 2744914
- Fabio Cavalletti and Martin Huesmann, Existence and uniqueness of optimal transport maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, 1367–1377. MR 3425266, DOI https://doi.org/10.1016/j.anihpc.2014.09.006
- Fabio Cavalletti and Andrea Mondino, Measure rigidity of Ricci curvature lower bounds, Adv. Math. 286 (2016), 430–480. MR 3415690, DOI https://doi.org/10.1016/j.aim.2015.09.016
- Fabio Cavalletti and Andrea Mondino, Optimal maps in essentially non-branching spaces, Commun. Contemp. Math. 19 (2017), no. 6, 1750007, 27. MR 3691502, DOI https://doi.org/10.1142/S0219199717500079
- Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
- Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161. MR 1440931, DOI https://doi.org/10.1007/BF02392620
- Nicola Gigli, Optimal maps in non branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal. 22 (2012), no. 4, 990–999. MR 2984123, DOI https://doi.org/10.1007/s00039-012-0176-5
- Nicola Gigli, Tapio Rajala, and Karl-Theodor Sturm, Optimal maps and exponentiation on finite-dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal. 26 (2016), no. 4, 2914–2929. MR 3544946, DOI https://doi.org/10.1007/s12220-015-9654-y
- Shouhei Honda, On low-dimensional Ricci limit spaces, Nagoya Math. J. 209 (2013), 1–22. MR 3032136, DOI https://doi.org/10.1017/S0027763000010667
- Martin Kell, Transport maps, non-branching sets of geodesics and measure rigidity, Adv. Math. 320 (2017), 520–573. MR 3709114, DOI https://doi.org/10.1016/j.aim.2017.09.003
- Christian Ketterer and Tapio Rajala, Failure of topological rigidity results for the measure contraction property, Potential Anal. 42 (2015), no. 3, 645–655. MR 3336992, DOI https://doi.org/10.1007/s11118-014-9450-5
- Yu Kitabeppu and Sajjad Lakzian, Characterization of low dimensional $RCD^*(K,N)$ spaces, Anal. Geom. Metr. Spaces 4 (2016), no. 1, 187–215. MR 3550295, DOI https://doi.org/10.1515/agms-2016-0007
- Stefano Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 85–120. MR 2267755, DOI https://doi.org/10.1007/s00526-006-0032-2
- John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI https://doi.org/10.4007/annals.2009.169.903
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
- Andrea Mondino and Aaron Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1809–1854. MR 3945743, DOI https://doi.org/10.4171/JEMS/874
- Shin-ichi Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), no. 4, 805–828. MR 2341840, DOI https://doi.org/10.4171/CMH/110
- Tapio Rajala and Karl-Theodor Sturm, Non-branching geodesics and optimal maps in strong $CD(K,\infty )$-spaces, Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 831–846. MR 3216835, DOI https://doi.org/10.1007/s00526-013-0657-x
- Timo Schultz, Existence of optimal transport maps in very strict $CD(K,\infty )$-spaces, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Paper No. 139, 11. MR 3846900, DOI https://doi.org/10.1007/s00526-018-1414-y
- Timo Schultz, Equivalent definitions of very strict $CD(K,N)$ -spaces, Preprint, arXiv:1906.07693 (2019).
- Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI https://doi.org/10.1007/s11511-006-0002-8
- Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. MR 2237207, DOI https://doi.org/10.1007/s11511-006-0003-7
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C23
Retrieve articles in all journals with MSC (2010): 53C23
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
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