Recent developments on metric measure spaces with Ricci curvature bounded below

Speaker:Prof. Shouhei Honda (Tohoku University)
Schedule:July 26(Tue), 27(Wed), 28(Thu), 10:00 - 11:30am
Venue:Zoom Meeting ID: 816 4977 5126; Passcode: Kahler

Description

In [13, 14, 15], Cheeger-Colding established the deep structure theory on Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature below. Moreover Jiang and Naber with them in [16, 17] proved further structure results on such spaces. On the other hand Cheeger-Colding asked in an appendix of [13] whether their theory can be covered by a synthetic way. Now we know the best answer to this question, namely RCD spaces give the best framework in a synthetic treatment of Ricci curvature lower bounds, in order to cover the theory on limit spaces as above. In the three lectures, we will introduce the basics, the techniques, and the recent results for RCD spaces. In particular we will focus on blow-up analysis on such spaces, which play significant roles in many situations. Finally I will provide open problems related to this topic.

Prerequisite
No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of Riemannian geometry.

References
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[3] L. Ambrosio, N. Gigli, G. Savare, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405--1490.
[4] L. Ambrosio, N. Gigli, G. Savare, Bakry-\'Emery curvature-dimension condition and Riemannian Ricci curvature bounds.Ann. of Prob. 43 (2015), 339--404.
[5] L. Ambrosio, S. Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, Measure theory in non-smooth spaces, 1–51, Partial Differ. Equ. Meas. Theory,De Gruyter Open, Warsaw, 2017.
[6] L. Ambrosio, S. Honda, Local spectral convergence in RCD^*(K, N) spaces. Nonlinear Anal. 177 Part A (2018), 1–23.
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[9] C. Brena, N. Gigli, S. Honda, X. Zhu, Weakly noncollapsed RCD spaces are strongly noncollapsed, arXiv:2110.02420
[10] E. Brue, E. Pasqualetto, D. Semola, Rectifiability of RCD(K,N) spaces via \delta-splitting maps, Ann. Fenn. Math. 46 (2021), no. 1,465-482.
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[24] N. Gigli, A. Mondino, G. Savare,Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3), 111 (2015),1071-1129.
[25] S. Honda, New differential operator and noncollapsed RCD spaces, Geom. Topol. 24 (2020), 2127-2148.
[26] S. Honda, Y. Peng, A note on the topological stability theorem from RCD spaces to Riemannian manifolds, arXiv:2202.06500
[27] A. Mondino, A. Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019),1809–-1854.
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[31] B. Wang, X. Zhao, Canonical diffeomorphisms of manifolds near spheres, arXiv:2109.14803.


Notes:

 Lecture 1.pdf   Lecture 2.pdf   Lecture 3.pdf