Minimal and Evolving Networks

This thesis is devoted to the study of minimal networks from both the static and the dynamic point of view and in particular we consider problems at the interface of geometry and analysis. In the first chapter we search for networks embedded in a given Riemannian surface with minimal length that satisfy some topological constraints, namely that one of being spines of the surface. Using standard techniques of the calculus of variation, we prove that such minimal networks exist for any closed Riemannian surfaces and then we focus on special cases (flat torus, hyperbolic surfaces) to obtain more information about their classification. In the second chapter we let the networks evolve according to the ``gradient flow" of the length. Intuitively this means that the curves which form the network evolves with normal velocity equal to the curvature. We consider solution in strong sense and in particular we discuss the short time existence and the singularity formation at the maximal time of existence, generalizing some results for the curve shortening flow of simple closed curves.