In the late '70s, motivated by the study of higher codimension minimal surfaces, F. Almgren introduced the notion of multiple valued functions minimizing the Dirichlet energy. As pointed out later by P. Mattila, Almgren's approach does not extend to the case of elliptic quadratic energies. In this talk I will present a complete characterization of lower semicontinuous functionals defined on Sobolev spaces of multiple valued functions, recovering earlier results by P. Mattila as special case and answering to one of his conjectures. This is a joint work with M. Focardi (Firenze) and C. De Lellis (Zuerich).

We consider hypersurfaces in 3-manifolds moving by an area preserving variant of the Willmore Flow, focusing on 'large' spheres in asymptotically Schwarzschild ambient metrics. We study short time existence and, largely relying on techniques used by Kuwert & Schätzle in the early '00s for Willmore flow in Euclidean Space, give generalizations of their observations concerning the minimal existence time and Blow-up-behaviour around singularities, adressing issues arising from the ambient curvature and from non-local terms in our flow equation. This is work in progress and, as part of an ongoing PhD thesis, contains a couple of rough edges.