Networks: existence of minimal configurations and curvature flow singularities

This dissertation revolves around the study of networks and the construction of configurations that locally minimise length. Inspired by a conjecture by Hass and Morgan concerning the existence of theta-shaped minimal networks in convex 2-spheres, and existence and non-existence results in planar convex sets by Freire, this work extends the inquiry to higher dimensions, providing new existence and multiplicity results. The methodology employed in proving these results is perturbative. We first identify a critical manifold, around which we apply a finite-dimensional reduction method coupled with the Lusternik–Schnirelmann category. Additionally, the dynamical aspect of networks is investigated through their curvature flow. Under specific symmetry conditions, we demonstrate that the number of singularities developed during the flow of networks with two triple junctions is finite, before a possible blow-up of the curvature. This finding enables the possibility of extending the flow indefinitely in certain cases, which in the limit would approach a (possibly degenerate) minimal network. A key aspect is the identification of a monotonous quantity, which is the number of intersections of the evolving network with a suitable stationary solution to the flow. This study delves into two ambient surfaces: the Euclidean plane and the standard round sphere. Analysing the long-time behaviour of the flow on the 2-sphere, we extend prior results on singularity analysis from the planar case to exclude the possibility of curvature blow-up without the length of a curve approaching zero. In particular, the multiplicity one conjecture is established in the case of symmetric networks on the 2-sphere.