Calculus of Variations and Geometric Measure Theory

F. Cavalletti - A. Mondino

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

created by cavallett on 22 Feb 2015
modified on 24 Oct 2016


Accepted Paper

Inserted: 22 feb 2015
Last Updated: 24 oct 2016

Journal: Invent. Math.
Year: 2015


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 $N \in [1,\infty)$, then the classic L\'evy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, 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 $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. 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 and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.