Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

By Fabio Cavalletti, Andrea Mondino. Inventiones Mathematicae, 2015

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We prove that if (X,d,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\mathsf {d},\mathfrak {m})$$\end{document} is a metric measure space with m(X)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {m}(X)=1$$\end{document} having (in a synthetic sense) Ricci curvature bounded from below by K>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K>0$$\end{document} and dimension bounded above by N∈[1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in [1,\infty )$$\end{document}, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any K∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathbb {R}$$\end{document}, N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 1$$\end{document} and upper diameter bounds) holds, i.e. the isoperimetric profile function of (X,d,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\mathsf {d},\mathfrak {m})$$\end{document} is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume v∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in (0,1)$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in (0,1)$$\end{document} 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/90d01dc241964e353d3790f69d9dc7e6948ff852

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Citations in the corpus

Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “1 Introduction In the last ten years there has been a great interest in the study of metric measure spaces with Ricci curvature bounded from below, see for instance [44], [55], [56], [29], [5], [6], [27], [7], [48], [49], [30], [24], [26], [38], [9], [47], [13], [12].”
The globalization theorem for the Curvature-Dimension condition (2016, methodology): “formation encoded in a family of suitably constructed L1-OptimalTransport rays, onto a given W2-geodesic. This goes in exactly the opposite direction to the one 7 studied by Cavalletti and Mondino in [27], and completes the cycle: CDloc(K,N) ⇒ CD1(K,N) ⇒ CD(K,N). To the best of our knowledge, this d”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “An extension of Milman’s results is due to Cavalletti and Mondino [13], who used the OMT-approach in combination to themeasure disintegrationmethod of Klartag [33] to prove a sharp Lévy-Gromov-type isoperimetric inequality on metric measure spaces satisfying the CD(κ, N ) condition with κ > 0 and N ”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, methodology): “It is then clear that the comparison and almost rigidity theorems for λ(1,1) will be based on the corresponding isoperimetric ones obtained by the authors in [17].”
Sobolev Inequalities in Manifolds with Nonnegative Curvature (2020, background): “The LévyGromov inequality was recently generalized in [13] and [8].”
CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES (2019, background): “Mondino have been able to extend in Cavalletti and Mondino [2017a,b] the localization technique to obtain in the class of essentially non-branching CD(K; N )m.m.s. this and many other inequalities with sharp constants. Levy–Gromov inequality: Cavalletti and Mondino [2017a] If m(X) = 1 and K > 0, the”