Inserted: 25 aug 2021
Last Updated: 12 jan 2022
Journal: Journal of Dynamical and Control Systems
For a complete Riemannian manifold with bounded geometry, we prove the existence of isoperimetric clusters and also the compactness theorem for sequence of clusters in a larger space obtained by adding finitely many limit manifolds at infinity. Moreover, we show that isoperimetric clusters are bounded. We introduce and prove the Holder continuity of the multi-isoperimetric profile which has been explored by Emanuel Milman and Joe Neeman with a Gaussian-weighted notion of perimeter. We yield a proof of classical existence theorem, e.g. in space forms, for isoperimetric cluster using the results presented here. The results in this work generalize previous works of Stefano Nardulli, Andrea Mondino, Frank Morgan, Matteo Galli and Manuel Ritor\'e from the classical Riemannian and sub-Riemannian isoperimetric problem to the context of Riemannian isoperimetric clusters and also Frank Morgan and Francesco Maggi works on the clusters theory in the Euclidean setting.