Free boundary minimal surfaces: a nonlocal approach

Abstract

Given a C-k-smooth closed embedded manifold N subset of R-m, with k >= 2, and a compact connected C-infinity-smooth Riemannian surface (S, g) with partial derivative S not equal empty set, we consider 1/2-harmonic maps u is an element of H-1/2(partial derivative S, N). These maps are critical points of the nonlocal energy E(f; g) := integral S vertical bar del(u) over tilde vertical bar(2) dvol(g), (0.1) where (u) over tilde is the harmonic extension of u in S. We express the energy (0.1) as a sum of the 1/2-energies at each boundary component of partial derivative S (suitably identified with the circle S-1), plus a quadratic term which is continuous in the H-s (S-1) topology, for any s is an element of R. We show the C-k-1,C-delta regularity of 1/2-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of E with respect to variations of the pair (f, g), in terms of the Teichmuller space of S.