Calculus of Variations and Geometric Measure Theory

F. Cavalletti - F. Maggi - A. Mondino

Quantitative isoperimetry à la Levy-Gromov

created by mondino on 13 Jul 2017
modified on 12 Jul 2018


Accepted Paper

Inserted: 13 jul 2017
Last Updated: 12 jul 2018

Journal: Comm. Pure and Applied Math.
Year: 2017


On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are actually obtained in the more general context of (possibly non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method.