The isoperimetric problemviadirect method in noncompact metric measure spaces with lower Ricci bounds

Abstract

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N) spaces (X, d, ℋN). 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 spaceXalong 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 setsEi⊂Xiwith uniformly bounded measure and perimeter, where (Xi,di, ℋN) is an arbitrary sequence of 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.