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Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
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Cavalletti, Fabio and Mondino, Andrea (2017) Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds. Inventiones Mathematicae, 208 (3). pp. 803-849. doi:10.1007/s00222-016-0700-6 ISSN 0020-9910.
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Official URL: http://dx.doi.org/10.1007/s00222-016-0700-6
Abstract
We prove that if (X, d,m) is a metric measure space with m(X) = 1
having (in a synthetic sense) Ricci curvature bounded from below by K > 0 and dimension bounded above by N ∈ [1,∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any K ∈ R, N ≥ 1 and upper diameter bounds) holds, i.e. the isoperimetric profile function of (X, d,m) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured
Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.
Item Type: | Journal Article | ||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Journal or Publication Title: | Inventiones Mathematicae | ||||||||||
Publisher: | Springer | ||||||||||
ISSN: | 0020-9910 | ||||||||||
Official Date: | 1 June 2017 | ||||||||||
Dates: |
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Volume: | 208 | ||||||||||
Number: | 3 | ||||||||||
Page Range: | pp. 803-849 | ||||||||||
DOI: | 10.1007/s00222-016-0700-6 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||
Date of first compliant deposit: | 24 October 2017 | ||||||||||
Date of first compliant Open Access: | 19 November 2017 |
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