The isoperimetric problem<i>via</i>direct method in noncompact metric measure spaces with lower Ricci bounds

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