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

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD( K, N ) spaces ( X , d, 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 ⊂ X i with uniformly bounded measure and perimeter, where ( X i , d i , ℋ N ) 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.