Inserted: 25 jun 2020
Last Updated: 17 jul 2020
This article is part of my Ph.D thesis written under the advising of Stefano Nardulli. I would like to give a special thanks to Stefano Nardulli, for his enthusiasm for the project, for his contribution with precious ideas and for bringing my attention to the subject of this paper. The discussions, encouragements and comments of my co-advisor, Glaucio Terra, were very valuable for this work. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 88882.3779542019-01.
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. As far as we know, we introduce for the first time in the literature the multi-isoperimetric profile and prove its H¨older continuity. 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, 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.
Keywords: existence of isoperimetric clusters, minimizing clusters, multi-isoperimetric problem