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

¹ Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
² Centro de Matemática Cognição Computação, Universidade Federal do ABC, Avenida dos Estados, 5001. Bairro Bangu - Santo André - SP - Brasil. CEP 09210-580
³ Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, via Cintia, Monte S. Angelo, 80126 Napoli, Italy

^* Corresponding author: gioacchino.antonelli@sns.it

Received: 19 January 2022
Accepted: 11 July 2022

Abstract

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.