On the topology and the boundary of N-dimensional RCD(K,N) spaces

Abstract

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.

arXiv comment32 pages. Final version to appear in "Geometry & Topology"

References in corpus (19)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Alexandrov meets Lott--Villani--Sturm
• Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• Sharp measure contraction property for generalized H-type Carnot groups
• Optimal maps in essentially non-branching spaces
• New formulas for the Laplacian of distance functions and applications
• Obata's rigidity theorem for metric measure spaces
• Measure contraction properties of Carnot groups
• On quotients of spaces with Ricci curvature bounded below
• New differential operator and non-collapsed $RCD$ spaces
• Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts
• Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below
• On the volume measure of non-smooth spaces with Ricci curvature bounded below
• Metric-measure boundary and geodesic flow on Alexandrov spaces
• Measure rigidity of Ricci curvature lower bounds
• An overview on the proof of the splitting theorem in non-smooth context
• Characterization of Low Dimensional $RC{D}^{\ast }(K,N)$ spaces

Cited by in corpus (16)

• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• New differential operator and non-collapsed $RCD$ spaces
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• Singular Weyl's law with Ricci curvature bounded below
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces
• An upper bound on the revised first Betti number and a torus stability result for RCD spaces
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• On the structure of RCD spaces with upper curvature bounds
• Isometric immersions of RCD$(K,N)$ spaces via heat kernels
• The metric measure boundary of spaces with Ricci curvature bounded below
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth
• On gluing Alexandrov spaces with lower Ricci curvature bounds
• Improved regularity estimates for Lagrangian flows on $\text{RCD}(K,N)$ spaces