Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

Kapovitch, Vitali; Ketterer, Christian

doi:10.1515/agms-2019-0010

Cited by 8 publications

(2 citation statements)

References 41 publications

(59 reference statements)

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“…1. (X, d) has an upper bound on sectional curvature in a synthetic sense, namely, it is a CAT(κ) space for some κ > 0: [47] 2. (X, d) is isometric to a smooth Riemannian manifold, possibly with boundary: [39].…”

Section: Main Results and Some Commentsmentioning

confidence: 99%

Weakly non-collapsed RCD spaces are strongly non-collapsed

Brena¹,

Gigli²,

Honda³

et al. 2021

Preprint

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We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality.One of the auxiliary results of independent interest that we obtain is about the link between the properties -tr(Hessf ) = ∆f on U ⊆ X for every f sufficiently regular, m = cH n on U ⊆ X for some c > 0, where U ⊆ X is open and X is a -possibly collapsed -RCD space of essential dimension n.

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Section: Main Results and Some Commentsmentioning

confidence: 99%

Weakly non-collapsed RCD spaces are strongly non-collapsed

Brena¹,

Gigli²,

Honda³

et al. 2021

Preprint

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“…Instead of the noncollapsing condition one can also add an upper curvature bound. This will also force the limit to become RCD by [KK19a,KK19b,KKK19] Corollary 1.14. Let {(M i , o i )} i∈N be a sequence of n-dimensional, pointed Riemannian manifolds that satisfy the condition CD(κ i , N )…”

Section: Introductionmentioning

confidence: 99%

Stability of metric measure spaces with integral Ricci curvature bounds

Ketterer¹

2020

Preprint

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In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of n-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition CD(K, n) in the sense of Lott-Sturm-Villani provided the L p -norm for p > n 2 of the part of the Ricci curvature that lies below K converges to 0. The results also hold for sequences of general smooth metric measure spaces (M, g M , e −f vol M ) where Bakry-Emery curvature replaces Ricci curvature. Corollaries are a Brunn-Minkowski-type inequality, a Bonnet-Myers estimate and a statement on finiteness of the fundamental group. Together with a uniform noncollapsing condition the limit even satisfies the Riemannian curvature-dimension condition RCD(K, N ). This implies volume and diameter almost rigidity theorems.

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On the structure of $$\mathrm {RCD}$$ spaces with upper curvature bounds

2022

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We develop a structure theory for $$\mathrm {RCD}$$ RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are $$\mathrm {DC}$$ DC coordinates and the distance is induced by a continuous $$\mathrm {BV}$$ BV Riemannian metric.

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Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

Cited by 8 publications

References 41 publications

Weakly non-collapsed RCD spaces are strongly non-collapsed

Weakly non-collapsed RCD spaces are strongly non-collapsed

Stability of metric measure spaces with integral Ricci curvature bounds

On the structure of $$\mathrm {RCD}$$ spaces with upper curvature bounds

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