Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows

By Nicola Gigli, Andrea Mondino, Giuseppe Savar'e. 2013

286 citations recorded by the source corpus.

The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition. We propose various definitions, and show that all of them are equivalent and that for doubling spaces these are also equivalent to the well‐known measured Gromov–Hausdorff convergence.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a7f8528e7c4d568a70654b30f8aee1cd0ac71460

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In Applied Mathematics

Citations in the corpus

New formulas for the Laplacian of distance functions and applications (2018, background): “The class of RCD∗(K,N) spaces includes the following remarkable subclasses: – Measured Gromov Hausdorff limits of Riemannian N -dimensional manifolds satisfying Ricci ≥ K, see [4, 34].”
CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES (2019, background): “Remarkably, Gromov’s reconstruction theorem Gromov [2007] (extended in Gigli, Mondino, and Savaré [2015] to spaces with infinite mass), characterizes the equivalence classes by the family of functionals”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “The following formulation is due to [26] and [7],which fits the pmG-convergence well.”
On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces (2013, background): “While this characterization is extremely useful in the proof of stability properties [3,4,23], in the proof of localization or globalization properties it suffers the same limitations described in Remark 2.”
Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “We emphasize that the choice of excluding null reference measures is customary in this research field, see for instance [34] and references therein for a discussion of this topic in relation to convergence of mm-structures.”