Isoperimetric problems and lower bounds on curvature

2025/09/23 by Gioacchino Antonelli, Antonelli, Gioacchino
#math.DG

paper · pdf · doi:10.48550/arxiv.2509.18618

Abstract

This note surveys some classical results and recent developments on the interplay between lower curvature bounds and the isoperimetric problem. It is based on mini-courses given at the "European Doctorate School of Differential Geometry" (Granada, July 2024) and the summer school "Optimal transport, heat flow and synthetic Ricci bounds" (Chicago, June 2025).

Citations

• The globalization theorem for the Curvature-Dimension condition
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• On the differential structure of metric measure spaces and applications
• Ricci curvature for metric-measure spaces via optimal transport
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Curvature estimates for minimal hypersurfaces
• An optimal Poincaré inequality for convex domains

Related

• Curvature bounds via an isoperimetric comparison for Ricci flow on\n surfaces
• Isoperimetric Bounds on Convex Manifolds
• More on functional and quantitative versions of the isoperimetric inequality
• Isoperimetric characterization of upper curvature bounds
• Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere