The rigidity of sharp spectral gap in nonnegatively curved spaces

Ketterer, Christian; Kitabeppu, Yu; Lakzian, Sajjad

doi:10.48550/arxiv.2110.05045

Cited by 3 publications

(5 citation statements)

References 70 publications

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“…Proof. The proof of the corollary is exactly the content of section 5 and section 6 in [KKL21] that results in the proof of Theorem 6.10 in [KKL21] that corresponds to our statement.…”

Section: Then We Consider Measurable Setsmentioning

confidence: 57%

“…Thanks to locality of Hessf for f ∈ H 2.2 (X) the Hessian Hess(u) for u ∈ H 2,2 loc (Ω) is well-defined. The following theorem is Corollary 4.16 in [KKL21].…”

Section: Then We Consider Measurable Setsmentioning

confidence: 91%

“…Remark 4.12. For the proof of the main theorem in [KKL21] the authors show that the induced intrinsic metric of Ω = f −1 ((− min f, max f )) splits off an interval where f = cos −1 •u with an eigenfunction u on a compact RCD(0, N ) space X.…”

Section: Then We Consider Measurable Setsmentioning

confidence: 99%

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

Ketterer¹

2021

Preprint

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We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results for Riemannian manifolds with boundary by Kasue, Croke and Kleiner to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove stability of minimal and more general CMC hypersurfaces and almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

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“…Proof. The proof of the corollary is exactly the content of section 5 and section 6 in [KKL21] that results in the proof of Theorem 6.10 in [KKL21] that corresponds to our statement.…”

Section: Then We Consider Measurable Setsmentioning

confidence: 57%

“…Thanks to locality of Hessf for f ∈ H 2.2 (X) the Hessian Hess(u) for u ∈ H 2,2 loc (Ω) is well-defined. The following theorem is Corollary 4.16 in [KKL21].…”

Section: Then We Consider Measurable Setsmentioning

confidence: 91%

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

Ketterer¹

2021

Preprint

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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: 58%

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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“…(3.5) The isomorphic splitting then follows from the recent [55, Theorem 1.4] and [56], extending the classical Riemannian result [53, Theorem C].…”

Section: Non Negatively Curved Spacesmentioning

confidence: 73%

Asymptotic isoperimetry 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 asymptotic isoperimetric properties for small and large volumes on N -dimensional RCD(K, N ) spaces (X, d, H N ). Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements are new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

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The rigidity of sharp spectral gap in nonnegatively curved spaces

Cited by 3 publications

References 70 publications

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

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

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

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