Isoperimetric inequality in noncompact MCP spaces

TL;DR

This paper establishes a sharp isoperimetric inequality for noncompact metric measure spaces with synthetic Ricci curvature bounds, using a novel localization scaling approach instead of classical methods.

Contribution

It introduces a new proof technique for isoperimetric inequalities in MCP spaces that bypasses the need for Brunn-Minkowski and PDE methods, applicable in singular settings.

Findings

Proves a sharp isoperimetric inequality for MCP(0,N) spaces

Develops a localization scaling method for non-smooth spaces

Extends isoperimetric results to singular metric measure spaces

Abstract

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 localisation.