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Contributions to the theory of free boundary minimal surfaces

In this thesis, we present various contributions to the study of free boundary minimal surfaces. After introducing some basic tools and discussing some delicate aspects related to the definition of Morse index when allowing for a contact set, we divide the thesis in two parts. In the first part of this dissertation, we study free boundary minimal surfaces with bounded Morse index in a three-dimensional ambient manifold. More specifically, we present a degeneration analysis of a sequence of such surfaces, proving that (up to subsequence) they converge smoothly away from finitely many points and that, around such `bad' points, we can at least `uniformly' control the topology and the area of the surfaces in question. As a corollary, we obtain a complete picture of the way different `complexity criteria' (in particular: topology, area and Morse index) compare for free boundary minimal surfaces in ambient manifolds with positive scalar curvature and mean convex boundary. In the second part, we focus on an equivariant min-max scheme to prove the existence of free boundary minimal surfaces with a prescribed topological type. The principle is to choose a suitable group of isometries of the ambient manifold in order to obtain exactly the topology we are looking for. We recall a proof of the equivariant min-max theorem, and we also prove a bound on the Morse index of the resulting surfaces. Finally, we apply this procedure to show the existence of a new family of free boundary minimal surfaces with connected boundary and arbitrary genus in the three-dimensional unit ball.