## – Europe/Lisbon

Room P3.10, Mathematics Building — Online

Small minimal clusters in Riemannian (or Finsler) manifolds

Since a Riemannian manifold is locally similar to the Euclidean space, it is easy to see that isoperimetric sets of small volume in such a manifold are very close to balls, and in particular they are connected. Much less is known for the case of minimal clusters. In this talk, we will describe the general situation and we will present a recent result showing that also small minimal clusters are connected if the ambient space is a compact Riemannian manifold. In addition, we will discuss also the situation for Finsler manifolds, showing that a small minimal m-cluster can have at most m connected components. While it might seem reasonable that also in this case small minimal clusters are connected, we will present an example showing that this is not true. We will conclude by listing some open problems. (Most of the presented results are based on joint works with D. Carazzato and S. Nardulli).