Paper · Mar 6, 2017

Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds

Fabio Cavalletti, Andrea Mondino

math.DGarXiv:1703.02119v2
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Abstract

Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

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Submitted
Mar 6, 2017
Last revised
Apr 18, 2018 (v2)
Subjects
math.DG, math.MG
Journal
Int. Math. Res. Not. IMRN, Vol 2020, no. 5, 1481--1510
DOI
10.1093/imrn/rny070

Comments: 22 pages. Final version published by Int. Math. Res. Not. (IMRN)

Ricci flowWeak & singular flowsLimit spaces & RCDGeometric inequalities

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On a isoperimetric-isodiametric inequality (Andrea Mondino and Emanuele Spadaro, 2016)3Mondino 2016 New Volume Comparison results and Applications to degeneration of Riemannian metrics (Qi S Zhang and Meng Zhu, 2016)6Zhang 2016 Existence and applications of Ricci flows via pseudolocality (Fei He, 2016)4He 2016 On quotients of spaces with Ricci curvature bounded below (Fernando Galaz-García, Martin Kell, Andrea Mondino et al., 2017)2Galaz-García 2017 Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds (Felix Schulze, 2018)8Schulze 2018 Pseudolocality theorems of Ricci flows on incomplete manifolds (Liang Cheng, 2022)1Cheng 2022 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)5Sturm 2024 Willmore-type inequalities for closed hypersurfaces in weighted manifolds (Guoqiang Wu and Jia-Yong Wu, 2024)7Wu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudolocality theorems of Ricci flows on incomplete manifolds

    Liang Cheng

    Topics Geometric inequalities · Shared terms “Euclidean isoperimetric”, “almost Euclidean”, “isoperimetric inequality”

  2. On quotients of spaces with Ricci curvature bounded below

    Fernando Galaz-García, Martin Kell, Andrea Mondino et al.

    Topics Limit spaces & RCD, Weak & singular flows · Same author Andrea Mondino · Shared terms “curvature lower”, “curvature bounded”, “lower bound”

  3. On a isoperimetric-isodiametric inequality

    Andrea Mondino and Emanuele Spadaro

    Topics Geometric inequalities · Same author Andrea Mondino · Shared terms “isoperimetric”, “metric ball”, “Euclidean”

  4. Existence and applications of Ricci flows via pseudolocality

    Fei He

    Topics Geometric inequalities · Shared terms “Euclidean isoperimetric”, “almost Euclidean”, “isoperimetric inequality”

  5. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm

    Topics Geometric inequalities, Limit spaces & RCD, Weak & singular flows · Shared terms “isoperimetric inequality”, “synthetic”

  6. New Volume Comparison results and Applications to degeneration of Riemannian metrics

    Qi S Zhang and Meng Zhu

    Topics Geometric inequalities, Limit spaces & RCD · Shared terms “isoperimetric inequality”

  7. Willmore-type inequalities for closed hypersurfaces in weighted manifolds

    Guoqiang Wu and Jia-Yong Wu

    Topics Geometric inequalities, Limit spaces & RCD · Shared terms “isoperimetric”

  8. Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

    Felix SchulzeWider flows

    Topics Geometric inequalities, Weak & singular flows · Shared terms “isoperimetric inequality”, “Euclidean isoperimetric”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.