Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

Ketterer, Christian

doi:10.48550/arxiv.2111.12020

Cited by 2 publications

(9 citation statements)

References 21 publications

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“…where the inequality follows from (6.1) applied on the isoperimetric set E t , since we already know that any mean curvature barrier for E t is nonnegative. Since t > 0 is arbitrary, we proved that ∆d E = 0 on X \ E. (6.5) The isomorphic splitting then follows from the recent [75,Theorem 1.4] and [76], extending the classical Riemannian result [71,Theorem C].…”

Section: The Case Of Nonnegatively Curved Spacesmentioning

confidence: 61%

“…Compact convex subsets of Riemannian manifolds with Ricci curvature lower bounds have been considered in [27]. • The stability of mean curvature barriers in the sense of Theorem 1.2 for Gromov-Hausdorff converging sequences of boundaries inside measured Gromov-Hausdorff converging sequences of RCD(K, N ) spaces has been recently observed in [75] (see also the previous [32]). The main novelty of our work in this regard is to provide a large and natural class of sets having mean curvature barriers, namely isoperimetric sets.…”

Section: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 99%

“…In this section we derive a stability result for sequences of isoperimetric sets converging in L 1 to a limit set, improving the L 1 convergence to Hausdorff convergence. First, we need the following technical lemma, independently observed also in the recent [75], relating convergence in Hausdorff sense and convergence of mean curvature barriers. Lemma 5.1.…”

Section: Stability Of Isoperimetric Setsmentioning

confidence: 99%

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Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N -dimensional RCD(K, N ) spaces (X, d, H N ). Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when K = 0 under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Contents 1. Introduction Isoperimetry and lower Ricci curvature bounds Main results Consequences and strategies of the proofs Sharp asymptotic behaviour for small and large volumes Comparison with the previous literature Acknowledgements 2. Preliminaries 2.1. Convergence and stability results 2.2. BV functions and sets of finite perimeter in metric measure spaces 2.3. Sobolev functions, Laplacians and vector fields in metric measure spaces 2.4. Geometric Analysis on RCD spaces 2.5. Localization of the Curvature-Dimension condition 3. The distance function from isoperimetric sets 4. Concavity properties of the isoperimetric profile function and consequences 4.1. Sharp concavity inequalities for the isoperimetric profile 4.2. Fine properties of the isoperimetric profile 4.3. Consequences 5. Stability of isoperimetric sets

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Section: The Case Of Nonnegatively Curved Spacesmentioning

confidence: 61%

Section: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 99%

Section: Stability Of Isoperimetric Setsmentioning

confidence: 99%

See 1 more Smart Citation

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The second author is partially supported by the INdAM -GNAMPA Project 2022 CUP_E55F22000270001 "Isoperimetric problems: variational and geometric aspects". The authors thank Christian Ketterer for useful discussions on the results in [70,69]. They also thank Elia Bruè, Mattia Fogagnolo, Andrea Mondino and Daniele Semola for many fruitful discussions on the topic of this work during various stages of the writing of this paper.…”

Section: Do There Exist Isoperimetric Regions For Every Volume? Does ...mentioning

confidence: 93%

“…Then for a CBB(0) space having at least a limit at infinity of the form R × K with K compact, it is possible to derive the asymptotic behavior of the Busemann function (Definition 3.2) of a given ray, see Theorem 3.9, and Proposition 3.7. Hence, exploiting also recent rigidity results from [69,70], we derive the following theorem on the asymptotic behavior of the sublevel sets of the Busemann function.…”

Section: Do There Exist Isoperimetric Regions For Every Volume? Does ...mentioning

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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Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

Cited by 2 publications

References 21 publications

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

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

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