Ricci curvature in dimension 2

Lytchak, Alexander; Stadler, Stephan

doi:10.48550/arxiv.1812.08225

Cited by 5 publications

(6 citation statements)

References 28 publications

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“…) is an RCD (1, 2) space. But then the result of Lytchak-Stadler [LS18] implies that (Σ, d0 ) is a two dimensional Alexandrov space of curvature at least 1. Lemma 8.3.…”

Section: Compactness Of Ricci-pinched Manifolds In Dimension Threementioning

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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“…) is an RCD (1, 2) space. But then the result of Lytchak-Stadler [LS18] implies that (Σ, d0 ) is a two dimensional Alexandrov space of curvature at least 1. Lemma 8.3.…”

Section: Compactness Of Ricci-pinched Manifolds In Dimension Threementioning

confidence: 99%

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

Deruelle¹,

Schulze²,

Simon³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We recall that if (X, d) is an N -dimensional CBB(0) metric space, then the triple (X, d, H N ) is an RCD(0, N ) metric measure space, see [99,57]. Moreover, (X, d) is a 2-dimensional CBB(0) metric space if and only if (X, d, H N ) is RCD(0, N ), see [79]. Analogous results hold for any lower bound on the curvature.…”

Section: Preliminariesmentioning

confidence: 99%

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Antonelli¹,

Pozzetta²

2023

Preprint

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In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces.In the previous setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov-Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray.Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and, in such generality, we characterize the geometric rigidity at the level of the isoperimetric profile.Specializing our study to the 2-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary.The proofs exploit an approach by direct method and the results in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.

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“…The case N = 2 could be treated with arguments analogous to those considered here, with the slight modifications due to the difference behaviour of the Green function. Notice also that the theory of non collapsed RCD(K, 2) metric measure spaces is very well understood, thanks to [LS18], where it is shown that they are Alexandrov spaces with curvature bounded from below.…”

Section: 2mentioning

confidence: 99%

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

Mondino¹,

Semola²

2021

Preprint

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The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K, N ) metric measure spaces):• We establish a new principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the p-Hopf-Lax semigroup, for general exponents p ∈ [1, ∞).• We develop an instrinsic viscosity theory of Laplacian bounds.• We prove Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting.• We initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter. The class of RCD(K, N ) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks.

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Ricci curvature in dimension 2

Abstract: We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.

Cited by 5 publications

References 28 publications

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

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

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

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