Isoperimetric inequality in noncompact MCP spaces

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds MCP(0, N) and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for MCP(0, N), and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localization.

[1]  Shin-ichi Ohta.  Needle decompositions and isoperimetric inequalities in Finsler geometry , 2015, 1506.05876.

[2]  M. Gromov,et al.  Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces , 1987 .

[3]  Miklós Simonovits,et al.  Isoperimetric problems for convex bodies and a localization lemma , 1995, Discret. Comput. Geom..

[4]  Florian Johne,et al.  Sobolev inequalities on manifolds with nonnegative Bakry-\'Emery Ricci curvature , 2021, 2103.08496.

[5]  A. Mondino,et al.  New formulas for the Laplacian of distance functions and applications , 2018, Analysis & PDE.

[6]  Bo'az Klartag,et al.  Needle decompositions in Riemannian geometry , 2014, 1408.6322.

[7]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces , 2006 .

[8]  Emanuel Milman.  Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition , 2011 .

[9]  A. Mondino,et al.  Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds , 2015, Inventiones mathematicae.

[10]  A. Kristály,et al.  Geometric inequalities on Heisenberg groups , 2016, 1605.06839.

[11]  T. Rajala.  Failure of the local-to-global property for CD(K,N) spaces , 2013, 1305.6436.

[12]  Emanuel Milman,et al.  The globalization theorem for the Curvature-Dimension condition , 2016, Inventiones mathematicae.

[13]  E. Milman.  The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds , 2019, Communications on Pure and Applied Mathematics.

[14]  S. Brendle.  Sobolev Inequalities in Manifolds with Nonnegative Curvature , 2020, Communications on Pure and Applied Mathematics.

[15]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[16]  S. Bianchini,et al.  The Monge Problem for Distance Cost in Geodesic Spaces , 2011, 1103.2796.

[17]  F. Cavalletti,et al.  Isoperimetric inequality under Measure-Contraction property , 2018, Journal of Functional Analysis.

[18]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[19]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces. II , 2006 .

[20]  Davide Barilari,et al.  Sub-Riemannian interpolation inequalities , 2017, Inventiones mathematicae.

[21]  V. Agostiniani,et al.  Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature , 2018, Inventiones mathematicae.

[22]  Simone Di Marino,et al.  Perimeter as relaxed Minkowski content in metric measure spaces , 2016, 1603.08412.

[23]  Karl-Theodor Sturm,et al.  Non-branching geodesics and optimal maps in strong C D ( K , ∞ ) -spaces , 2013 .

[24]  S. Brendle.  The isoperimetric inequality for a minimal submanifold in Euclidean space , 2019, 1907.09446.

[25]  Shin-ichi Ohta.  On the measure contraction property of metric measure spaces , 2007 .

[26]  A. Mondino,et al.  Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications , 2020, 2004.08934.