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

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.

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

References in corpus (3)

• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Optimal maps in essentially non-branching spaces

Cited by in corpus (4)

• New formulas for the Laplacian of distance functions and applications
• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature