Free boundary minimal surfaces: a nonlocal approach

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$,with$k\ge 2$,andacompactconnectedsmoothRiemanniansurface$(S,g)$with$\partial S\neq\emptyset$,weconsider$\frac 12$-harmonicmaps$u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big|\nabla\widetilde u\big|^2,d$\text{vol}$_g,\end{equation} where $\tilde{u}$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$-energiesateachboundarycomponentof$\partial S$ (suitably identified with the circle ${\mathcal{S}}^1$), plus a quadratic term which is continuous in the $H^s({\mathcal{S}}^1)$ topology, for any $s\in \mathbb{R}$. We show the $C^{k-1,\delta }$ regularity of $\frac{1}{2}$-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 Teichm"uller space of $S$.