Isoperimetric inequality in noncompact 𝖬𝖢𝖯 spaces

Abstract

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds Measure Contraction property ( M C P ( 0 , N ) \mathsf {MCP}(0,N) ) and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for M C P ( 0 , N ) \mathsf {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 localization.