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 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 Ei ⊂ Xi with 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.

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

“…Moreover (2) in Theorem 1.4 recovers the result known on 2-dimensional Riemannian manifolds from [102], and (a) in Theorem 1.4 recovers the topological conclusion in [102,Theorem 1.1]. We remark that (1), (2) in Theorem 1.4 here are proved by applying the above mentioned direct method from [19,20], thus unifying these results as corollaries of the method.…”

“…Let us now recall a couple of results. The mass decomposition result in Theorem 2.22 is proved in [20] building on top of [91,19,21]. The result in Corollary 2.23 can be found in [23,Corollary 4.17].…”

“…Eventually, Theorem 1.1, Theorem 1.3 and Theorem 1.4 follow by variational arguments exploiting Theorem 1.7 and the approach by direct method developed in [19,20].…”

“…In the last fifty years, a great interest has been devoted to the study of several aspects of the isoperimetric problem on spaces with (classical or synthetic) notions of lower bounds on the curvature. We mention [32,31,27,59,89,83,84,41] concerning sharp, rigid and quantitative isoperimetric inequalities on compact spaces with Ricci lower bounds, [1,34,26,63,22,42,43] about sharp and rigid isoperimetric inequalities on noncompact spaces with nonnegative Ricci curvature and Euclidean volume growth, see (1.1) below, [27,113,89,73,28,29,30,86,23] for what concerns differential properties of the isoperimetric profile of spaces with Ricci lower bounds, and [102,104,90,91,86,49,19,18,20] studying existence of isoperimetric sets on noncompact spaces with lower curvature bounds. Specializing to the case of convex bodies in the Euclidean space, a number of results about existence of minimizers, isoperimetric inequalities, stability and topology of isoperimetric sets have been obtained in [77,113,106,74].…”

“…The link between the isoperimetric problem and the structure at infinity, together with recent developments on the Geometric Analysis on metric measure spaces with curvature bounded below (Section 2.3), has led to several contributions which exploit the direct method of the Calculus of Variations in order to find an isoperimetric region. Building on the fundamental concentration-compactness principle [76] and the results in [104,91,86], such direct method has been established in [19,20], see also [92,100,93] for related applications, leading to new existence results of minimizers under assumptions on the asymptotic structure of the ambient. In particular, in [18] it is considered the case of Riemannian manifolds with nonnegative Ricci curvature with Euclidean volume growth, i.e., such that AVR(X) := lim r→+∞ H N (B r (o))…”