The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

TL;DR

This paper develops a structure theorem for isoperimetric minimizers in noncompact RCD(K,N) spaces, identifying their limits via Gromov-Hausdorff convergence and establishing a generalized compactness result.

Contribution

It introduces a new generalized compactness theorem for sequences of sets in RCD spaces and characterizes the limits of isoperimetric minimizing sequences in noncompact metric measure spaces.

Findings

Limit of minimizing sequences identified by finite isoperimetric regions

Number of regions bounded linearly by measure of sequence

New criterion for convergence without mass loss at infinity

Abstract

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets $E_i\subset X_i$ with uniformly bounded measure and perimeter, where $(X_i,\mathsf{d}_i,\mathcal{H}^N)$ is an arbitrary sequence of $\mathsf{RCD}(K,N)$ spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.