Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem
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Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem ˜ Flores1 · Stefano Nardulli2 Abraham Enrique Munoz Received: 28 June 2020 / Revised: 17 September 2020 / Accepted: 19 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 C 2,α locally asymptotic bounded geometry with C 0 locally asymptotic bounded geometry. Keywords Existence of isoperimetric region · Isoperimetric profile Mathematics Subject Classification (2010) 49Q20 · 58E99 · 53A10 · 49Q05
1 Introduction In this paper, we study compactness of the family of finite perimeter sets with bounded volume and bounded perimeter, but with a possibly unbounded diameter in a complete noncompact Riemannian manifold, assuming some bounded geometry conditions on the ambient manifold. The difficulty is that for a sequence of regions, some volume may disappear to infinity. Given a sequence of finite perimeter sets inside a manifold with bounded geometry, we show that up to a subsequence the original sequence splits into an at most countable number of pieces which carry a positive fraction of the volume, one of them possibly staying at finite distance and the others concentrating along different directions. Stefano Nardulli
[email protected] Abraham Mu˜noz Flores [email protected] 1
Instituto de Matem´atica e Estat´ıstica, UERJ-Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
2
Centro de Matem´atica Cognic¸a˜ o e Computac¸a˜ o, UFABC-Universidade Federal do ABC, Santo Andr´e, SP, Brazil
˜ Flores and Stefano Nardulli Abraham Enrique Munoz
Moreover, each of these pieces will converge to a finite perimeter set lying in some pointed limit manifold, possibly different from the original. So a limit finite perimeter set exists in a generalized multipointed convergence. The range of applications of these results is wide. The vague notions invoked in this introductory paragraph will be made clear and rigorous in what follows.
1.1 Finite Perimeter Sets and Compactness In the remaining part of this paper, we always assume that all the Riemannian manifolds M considered are smooth with smooth Riemannian metric g. We denote by Vg the canonical Riemannian measure induced on M by g and by Ag the (n − 1)-Hausdorff measure associated with the canonical Riemannian length space metric d of M. When it is already clear from the context, explicit mention of the metric g will be suppressed in what follows. Definition 1.1 Let M be a Riemannian manifold of dimension n,
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