# Stability of RCD condition under concentration topology

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## Abstract

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 Classification

53C23## Notes

### Acknowledgements

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).

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