Moduli spaces of compact \(\mathrm{RCD}(0,N)\)-structures (Q6084202)

In mathematics, a \((0,N)\)-Ricci-curvature-dimension (the RCD\((0,N)\)-mm-spaces for short) and the \((0,N)\)-curvature-dimension condition (the CD\((0,N)\)-mm-space for short, \(N\in[1,\infty]\)) have had an important role in several classical problems of the calculus of variations, geometric measure theory and mathematical physics. Several authors studied the geometry of RCD\((0,N)\)-mm-spaces [\textit{R. Abraham} et al., Electron. J. Probab. 18, No. 14, 1--21 (2013; Zbl 1285.60004); \textit{L. Ambrosio} et al., Duke Math. J. 163, No. 7, 1405--1490 (2014; Zbl 1304.35310); Ann. Probab. 43, No. 1, 339--404 (2015; Zbl 1307.49044); \textit{K. Bacher} and \textit{K.-T. Sturm}, J. Funct. Anal. 259, No. 1, 28--56 (2010; Zbl 1196.53027); J. Funct. Anal. 275, 793--829 (2018; Zbl 1419.58020); Acta Math. 196, No. 1, 65--131 (2006; Zbl 1105.53035); \textit{F. Cavalletti} and \textit{K.-T. Sturm}, J. Funct. Anal. 262, 5110--5127 (2017; Zbl 1375.53053); \textit{K.-T. Sturm}, Acta Math. 196, No. 1, 133--177 (2006; Zbl 1106.53032); \textit{F. Cavalletti} and \textit{A. Mondino}, Invent. Math. 208, 803--849 (2012; Zbl 1244.53050); \textit{G. Wei} and \textit{W. Wylie}, J. Differ. Geom. 83, No. 2, 377--405 (2009; Zbl 1189.53036); \textit{P. W. Y. Lee} et al., Discrete Contin. Dyn. Syst. 36, No. 1, 303--321 (2016; Zbl 1326.53043); \textit{J. Lott} and \textit{C. Villani}, Ann. Math. (2), 169, No. 3, 903--991 (2009; Zbl 1178.53038); \textit{M. Erbar} et al., Invent. Math. 201, 993--1071 (2015; Zbl 1329.53059); \textit{L. Rifford}, Math. Control Relat. Fields 3, No. 4, 467--487 (2013; Zbl 1275.53034); \textit{N. Garofalo} and \textit{A. Mondino}, Nonlinear Anal. 95, 721--734 (2014; Zbl 1286.58016); \textit{A. Mondino} and \textit{A. Naber}, J. Eur. Math. Soc. 21, 1809--1854 (2019; Zbl 1468.53039); \textit{L. Rizzi}, Calc. Var. Partial Differ. Equ. 55(3), Paper No. 60, 20 p. (2016; Zbl 1352.53026); \textit{F. Baudoin} and \textit{N. Garofalo}, J. Eur. Math. Soc. (JEMS) 19, No. 1, 151--219 (2017; Zbl 1359.53018); \textit{D. Barilari} and \textit{L. Rizzi}, Commun. Contemp. Math. 20, No. 6, Article ID 1750081, 24 p. (2018; Zbl 1398.53038)]. The principal objective in this paper is to set the foundations and prove some topological results about moduli spaces of non-smooth structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space. The authors also introduce the notions of lift and push-forward of an RCD\((0, N)-\)structure, and present concisely the Albanese map and the soul map associated to a compact topological space admitting an RCD\((0, N)-\)structure.