Stability of RCD condition under concentration topology
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We prove the stability of the Riemannian curvature dimension condition introduced by Ambrosio–Gigli–Savaré under the concentration of metric measure spaces introduced by Gromov. This is an analogue of the result of Funano–Shioya for the curvature dimension condition of Lott–Villani and Sturm. These conditions are synthetic lower Ricci curvature bound for metric measure spaces. En route, we also prove the convergence of the Cheeger energy in our setting.
Mathematics Subject Classification53C23
This work started during the first author’s stay at Bonn University. He also would like to express his gratitude to Professor Karl-Theodor Sturm for his hospitality during his stay in Bonn. The authors thank Kohei Suzuki and the anonymous referee for their thorough reading and comments. The first author was partly supported by the Grant-in-Aid for JSPS Fellows (No.17J03507) and the second author was partly supported by JSPS KAKENHI (No.26800035).
- 11.Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser Boston, Inc., Boston (1999)Google Scholar
- 14.Kazukawa, D., Ozawa, R., Suzuki, N.: Stabilities of rough curvature dimension conditions. J. Math. Soc. Jpn. (To appear) Google Scholar
- 22.Shioya, T.: Metric measure limits of spheres and complex projective spaces. In: Gigli, N. (ed.) Measure Theory in Non-smooth Spaces, pp. 261–287. De Gruyter Open, Warsaw (2017) Google Scholar
- 26.Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)Google Scholar