On Clusters and the Multi-isoperimetric Profile in Riemannian Manifolds with Bounded Geometry

Abstract

For a complete Riemannian manifold with bounded geometry, we prove a generalized compactness theorem for sequences of clusters (with uniformly bounded perimeter and volume) in a larger space obtained by adding at most countable many limit manifolds at infinity, in the spirit of Muñoz Flores and Nardulli (Journal of Dynamical and Control Systems 1–11, 2020). The arguments presented in the proof of this generalized compactness theorem when applied to minimizing sequences of clusters give a generalized existence theorem for isoperimetric clusters in a larger space obtained by adding, in this case, only finitely many limit manifolds at infinity, as in Nardulli (Asian J Math 18(1):1–28, 2014). To achieve this goal, we show that isoperimetric clusters are bounded and also we prove the continuity of the multi-isoperimetric profile. In fact, we prove a stronger continuity property that is the Hölder continuity of the multi-isoperimetric profile. The multi-isoperimetric profile has been introduced recently in Milman and Neeman (2018) in the context of smooth metric measured spaces with a Gaussian-weighted notion of perimeter. This work generalizes to the context of Riemannian isoperimetric clusters some previous results about the classical Riemannian and sub-Riemannian isoperimetric problem, see Galli and Ritoré (Jour Math Anal Applic, 2012), Morgan (Trans Amer Math Soc 355(12), 2003), Mondino and Nardulli (Commun Anal Geom, 24(1):115–138, 2016), Nardulli (Glob Anal Geom, 36(2):111–131, 2009), Nardulli (Asian J Math 18(1):1–28, 2009), and Nardulli Bull Braz Math Soc (N.S.) 49(2):199–260, 2018), as well as results from clusters theory in the Euclidean setting, see Maggi (2012) and Morgan (Math Ann 299:697–714, 1994). In particular, as a consequence of our generalized existence results, we prove an existence theorem (in the classical sense) for isoperimetric clusters in a quite large class of noncompact Riemannian manifolds (the same considered in Mondino and Nadulli (Commun Anal Geom 24(1):115–138, 2016)) which includes, for instance, the space forms.