Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows
Elia Bruè,Daniele Semola +1 more
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In this paper, a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces was proved for a newly defined quasi-metric built from the Green function of the Laplacian.Abstract:
We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.read more
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An optimal transport formulation of the Einstein equations of general relativity
Andrea Mondino,Stefan Suhr +1 more
TL;DR: In this paper, an optimal transport formulation of the full Einstein equations of general relativity, linking the Ricci curvature of a space-time with the cosmological constant and the energy-momentum tensor, is given.
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On the topology and the boundary of N-dimensional RCD(K,N) spaces
Vitali Kapovitch,Andrea Mondino +1 more
TL;DR: 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.
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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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Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves
TL;DR: In this paper, Modena and Szekelyhidi introduced a new class of asymmetric Lusin-Lipschitz inequalities to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna-Lions theory.
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Rigidity of the 1-Bakry–Émery Inequality and Sets of Finite Perimeter in RCD Spaces
TL;DR: In this paper, the authors studied the asymptotic behavior of sets of finite perimeter over metric measure spaces and provided a characterization of the class of spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry-Emery inequality.
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Journal ArticleDOI
Ordinary differential equations, transport theory and Sobolev spaces.
TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
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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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Ricci curvature for metric-measure spaces via optimal transport
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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Differentiability of Lipschitz Functions on Metric Measure Spaces
TL;DR: In this paper, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
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On the structure of spaces with Ricci curvature bounded below. I
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.