The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

Antonelli, Gioacchino; Nardulli, Stefano; Pozzetta, Marco

doi:10.48550/arxiv.2201.03525

Cited by 2 publications

(7 citation statements)

References 47 publications

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“…The ability to deduce information about the isoperimetric behaviour of an RCD(K, N ) metric measure space (X, d, H N ) from Theorem 1.2 seems to be related to the existence of isoperimetric regions, which is not guaranteed, when H N (X) = ∞. We overcome this issue thanks to asymptotic mass decomposition result recently proved in [19], extending the previous [105,96,90,18] and building on a concentration-compactness argument. If (X, d) is compact, a minimizing sequence for the isoperimetric problem (1.3) for volume V > 0 converges, up to subsequences, to an isoperimetric set, by lower semicontinuity of the perimeter.…”

Section: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 95%

“…The two statements below are proved in [19] building on top of [96,18,20]. They will be key ingredients for the proof of Theorem 4.4.…”

Section: Concavity Properties Of the Isoperimetric Profile Function A...mentioning

confidence: 97%

“…This is not true, in general, when (X, d) is not compact as elementary examples illustrate. However, [19] shows that, if (X, d, H N ) is a non compact RCD(K, N ) space with H N (B 1 (p)) > v 0 for any p ∈ X for some v 0 > 0, then, up to subsequences, every minimizing sequence for the isoperimetric problem at a given volume V > 0 splits into finitely many pieces. One of them converges to an isoperimetric region in X, possibly of volume less than V (possibly zero).…”

Section: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 99%

“…The second conclusion can be reached adapting [95, Theorem 2], which shows that I is locally (1 − 1 N )-Hölder, cf. [19].…”

Section: Concavity Properties Of the Isoperimetric Profile Function A...mentioning

confidence: 99%

“…The argument in the proof Corollary 4.17 can be employed in order to prove a upper bound on the number N in Theorem 4.1 in terms of the volume V of the minimizing sequence. More precisely, in the hypotheses and notation of Theorem 4.1, one can show that there exists ε = ε(K, N, v 0 ) > 0 such that N ≤ 1 + V /ε, see [19].…”

Section: Corollary 417 Letmentioning

confidence: 99%

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Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

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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: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 95%

“…The two statements below are proved in [19] building on top of [96,18,20]. They will be key ingredients for the proof of Theorem 4.4.…”

Section: Concavity Properties Of the Isoperimetric Profile Function A...mentioning

confidence: 97%

Section: Isoperimetry and Lower Ricci Curvature Boundsmentioning

confidence: 99%

“…The second conclusion can be reached adapting [95, Theorem 2], which shows that I is locally (1 − 1 N )-Hölder, cf. [19].…”

Section: Concavity Properties Of the Isoperimetric Profile Function A...mentioning

confidence: 99%

Section: Corollary 417 Letmentioning

confidence: 99%

See 3 more Smart Citations

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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 isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

Cited by 2 publications

References 47 publications

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci 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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