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

Author(s)

Antonelli, Gioacchino ; Nardulli, Stefano ; Pozzetta, Marco

• Document type: Journal article
• Source: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS; v. 28, p. 32-pg., 2022-08-12.

Abstract

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N) spaces (X, d, Script capital 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 subset of X-i with uniformly bounded measure and perimeter, where (X-i, d(i), Script capital 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. (AU)

• FAPESP's process: 21/05256-0 - Geometric variational problems: existence, regularity and geometrical characterization of the solutions
• Grantee: Stefano Nardulli
• Support Opportunities: Research Grants - Young Investigators Grants