On quotients of spaces with Ricci curvature bounded below

Abstract

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of ${\mathsf{RCD}}^{\ast }(K,N)$-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An ${\mathsf{RCD}}^{\ast }(K,N)$-space is a metric measure space with an upper dimension bound $N$ and weighted Ricci curvature bounded below by $K$ in a generalized sense. In particular, this shows that if $(M,g)$ has Ricci curvature bounded below by $K\in \mathbb{R}$ and dimension $N$, then the quotient space is an ${\mathsf{RCD}}^{\ast }(K,N)$-space. Additionally, we tackle the same problem for the $\mathsf{CD}/{\mathsf{CD}}^{\ast }$ and $\mathsf{MCP}$ curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to ${\mathsf{RCD}}^{\ast }(K,N)$-spaces with essential minimal dimension $n\le N$; a structure theorem for ${\mathsf{RCD}}^{\ast }(K,N)$-spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows.

51 pages. Comments welcome!

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Cited by in corpus (17)

• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Structure of Submetries
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• On fundamental groups of RCD spaces
• Duality and quotient spaces of generalized Wasserstein spaces
• Convergence of Non-Symmetric Diffusion Processes on RCD spaces
• Geometric properties of disintegration of measures
• A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion equation
• Cheeger bounds on spin-two fields
• Projections in Lipschitz-free spaces induced by group actions
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
• Yamabe problem in the presence of singular Riemannian Foliations
• Curvature bound of Dyson Brownian Motion
• Finite flat spaces
• Characterization of foliations via disintegration maps
• Isometric Rigidity of compact Wasserstein spaces