On the topology and the boundary of N–dimensional RCD(K,N) spaces
Vitali Kapovitch,Andrea Mondino +1 more
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In this paper, the authors established topological regularity and stability of N-dimensional RCD(K,N) spaces up to a small singular set and introduced the notion of a boundary of such spaces and studied its properties, including its behavior under Gromov-Hausdorff convergence.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.read more
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Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces.
TL;DR: In this article, a Gauss-Green integration by parts formula tailored to the setting of sets of finite perimeter over RCD$(K,N) metric measure spaces is presented.
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Boundary regularity and stability for spaces with Ricci bounded below
TL;DR: In this paper, the authors studied the structure and stability of boundaries in noncollapsed RCD spaces, that is, metric-measure spaces with lower Ricci curvature bounded below.
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Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces
TL;DR: In this paper, the authors generalize the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in \cite{CheegerNaber13a}.
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Quantitative estimates for the Bakry-Ledoux isoperimetric inequality
Cong Hung Mai,Shin-ichi Ohta +1 more
TL;DR: In this article, a quantitative isoperimetric inequality for weighted Riemannian manifolds was established, which is the first quantitative inequality on non-compact spaces besides Euclidean and Gaussian spaces.
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On the structure of RCD spaces with upper curvature bounds
TL;DR: In this article, a structure theory for RCD spaces with curvature bounded above in Alexandrov sense is developed, and it is shown that any such space is a topological manifold with boundary whose interior is equal to the set of regular points.
References
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Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
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On the geometry of metric measure spaces. II
TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
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John Lott,Cédric Villani +1 more
TL;DR: In this paper, a notion of a length space X having nonnegative N-Ricci curvature, for N 2 [1;1], or having 1-RICci curvatures bounded below by K, for K2 R, was given.
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Jeff Cheeger,Tobias H. Colding +1 more
TL;DR: In this article, the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound was studied.
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