Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

Honda, Shinya

doi:10.3842/sigma.2020.021

Cited by 2 publications

(2 citation statements)

References 55 publications

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“…−1 dV ḡi ), is already an RCD(m − 1, m) space by the standard theory, it is not immediate that the metric limit (W ∞ , d ∞ ), when equipped with the k-dimensional Hausdorff measure H k , is an RCD(0, k) space. Before proving the claim, let us briefly recall the definition of RCD(0, k) spaces proposed in [1]; see also [19,27]. For a complete, Hausdorff and separable metric measure space (X, d X , m) the Cheeger energy is defined for any f ∈ L 2 (X) as…”

Section: Initial Data Locally Collapsing To Orbifold Model Spacesmentioning

confidence: 99%

Ricci flow smoothing for locally collapsing manifolds

Huang¹,

Wang²

2020

Preprint

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We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.

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Section: Initial Data Locally Collapsing To Orbifold Model Spacesmentioning

confidence: 99%

Ricci flow smoothing for locally collapsing manifolds

Huang¹,

Wang²

2020

Preprint

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“…We mention that it is today an open problem to understand whether an RCD(K, M) space (X, d, H N ) with M > N is, in fact, an RCD(K, N) space, cf. [38,Conjecture 4.2].…”

Section: Introductionmentioning

confidence: 99%

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

Antonelli,

Pasqualetto,

Pozzetta

2021

Preprint

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In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in RCD spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study.We prove that on an RCD(K, N ) space (X, d, H N ), with K ∈ R, N ≥ 2, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with (N − 1)-Ahlfors regular topological boundary coinciding with the essential boundary.The proof is based on a new Deformation Lemma for sets of finite perimeter in RCD(K, N ) spaces (X, d, m) and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters.The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.

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Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

Abstract: In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.

Cited by 2 publications

References 55 publications

Ricci flow smoothing for locally collapsing manifolds

Ricci flow smoothing for locally collapsing manifolds

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

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