Free boundary minimal surfaces: a nonlocal approach

Given a smooth closed embedded manifold N and a compact connected smooth Riemannian surface (S, g) with boundary, we consider half-harmonic maps from the boundary of S to N. These maps are critical points of the nonlocal energy given by the Dirichlet energy of the harmonic extension of u in S. We express this energy as a sum of the half-energies at each boundary component, plus a quadratic term which is continuous in the smooth topology. We show the regularity of half-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of E with respect to variations of the pair (map, metric), in terms of the Teichmüller space of S.