Curvature tensor of smoothable Alexandrov spaces

Lebedeva, Nina; Petrunin, Anton

doi:10.48550/arxiv.2202.13420

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…We note that a quadratic lower curvature decay in (1.2), combined with (1.1) is sufficient to guarantee that the Ricci curvature of (M n , g) weakly converges to 0 in the almost radial directions when we consider any blow-down, see [41] for the precise statement. However, even under the stronger two-sided quadratic curvature decay assumption, this weak convergence does not localize to (all) boundaries of large isoperimetric sets.…”

Section: Strict Stability For Many Volumesmentioning

confidence: 99%

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

et al. 2022

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In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

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Section: Strict Stability For Many Volumesmentioning

confidence: 99%

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

et al. 2022

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“…If one further assumes that the sectional curvature of the initial space is bounded from below by − A r 2 with r being the distance, and one blows down the solution, one obtains a solution having the same properties, in some weak sense, coming out of any asymptotic cone of (M 3 , g 0 ). Based on results of Lebedeva-Petrunin [LP22] using methods of Alexandrov geometry, it is then a static problem to show that such asymptotic cones must be flat which leads to a contradiction. This is where our approach differs, as we will now explain.…”

Section: Avr(g(t))mentioning

confidence: 99%

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

Deruelle¹,

Schulze²,

Simon³

2022

Preprint

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This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by c • t −1 converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. Contents1. Introduction 1.1. Overview 1.2. Outline of paper 1.3. Notation 1.4. Acknowldgements 2. Curvature decay 3. Existence of an adjustment map 4. Rough convergence rate 5. Polynomial convergence rate 6. Exponential convergence rate 7. Almost Ricci-pinched expanding gradient Ricci solitons 8. Compactness of Ricci-pinched manifolds in dimension three Appendix A. Volume and distance convergence, for solutions to Ricci flow with Ric ≥ −1 and curvature bounded by c 0 /t Appendix B. A splitting theorem of R. Hochard for the Ricci flow in a singular setting References

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Curvature tensor of smoothable Alexandrov spaces

Cited by 2 publications

References 16 publications

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

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