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

Gigli, Nicola; Mondino, Andrea; Savaré, Giuseppe

doi:10.1112/plms/pdv047

Cited by 170 publications

(358 citation statements)

References 142 publications

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“…This conjecture was proved in [ChC00b] by Cheeger-Colding and is extended to the case of RCD-spaces in [GMS13] by Gigli-Mondino-Savaré. Moreover similar spectral convergence of other differential operators acting on functions, which include the Schrödinger operator, the p-Laplacian, and the ∂-Laplacian, are also known.…”

Section: Introductionmentioning

confidence: 97%

Spectral convergence under bounded Ricci curvature

Honda

2017

Journal of Functional Analysis

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Abstract. For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. Contents

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Section: Introductionmentioning

confidence: 97%

Spectral convergence under bounded Ricci curvature

Honda

2017

Journal of Functional Analysis

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“…This convergence requires pGHconvergence of the underlying metric spaces and additionally weak convergence of the pushforward measures under the n -Gromov-Hausdorf approximations to the reference measure of the limit space. An comprehensive reference for this topic is [15].…”

Section: Remark 22 (Topology On Iso M (X))mentioning

confidence: 99%

The Isometry Group of an RCD ∗ Space is Lie

Sosa

2017

Potential Anal

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We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD * -spaces and, under extra assumptions, by CD-spaces, CD * -spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.

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“…[20] for details). Next, following [20], we recall various notions of convergence of functions de ned on p-mGH converging spaces.…”

Section: Pointed Measured Gromov-hausdor Convergence and Convergencmentioning

confidence: 99%

“…It is obvious that this is in fact a notion of convergence for isomorphism classes of p.m.m.s., moreover it is induced by a metric (see e.g. [20] for details). Next, following [20], we recall various notions of convergence of functions de ned on p-mGH converging spaces.…”

Section: Pointed Measured Gromov-hausdor Convergence and Convergencmentioning

confidence: 99%

“…x the tangent cone is unique and euclidean [21,27]. From the technical point of view we also make use of the ne convergence results for Sobolev functions proved in [4,20], and we prove estimates on harmonic approximations of distance functions (see in particular Proposition 4.3). Harmonic approximations of distance functions are well known for smooth Riemannian manifolds with lower Ricci curvature bounds, and are indeed one of the key technical tools in the Cheeger-Colding theory of Ricci limit spaces [12][13][14]; on the other hand for non-smooth RCD * (K, N)-spaces it seems they have not yet appeared in the literature, and we expect them to be a useful technical tool in the future development of the eld.…”

Section: Introductionmentioning

confidence: 99%

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Angles between Curves in Metric Measure Spaces

Han

Mondino

2017

Analysis and Geometry in Metric Spaces

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Abstract:The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD * (K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

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

Cited by 170 publications

References 142 publications

Spectral convergence under bounded Ricci curvature

Spectral convergence under bounded Ricci curvature

The Isometry Group of an RCD ∗ Space is Lie

Angles between Curves in Metric Measure Spaces

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