Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem

Abstract

For a complete noncompact Riemannian manifold with bounded geometry, we prove a “generalized” compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extend previous results contained in Nardulli (Asian J Math 18(1):1–28, 2014), in such a way that the main theorem is a generalization of the generalized existence theorem, i.e., Theorem 1 of Nardulli (Asian J Math 18(1):1–28, 2014). We replace C2,α locally asymptotic bounded geometry with C0 locally asymptotic bounded geometry.