Free boundary minimal surfaces: a nonlocal approach

Abstract

Given a $C^k$-smooth closed embedded manifold $\mathcal{N}\subset {\mathbb{R}}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\ne \varnothing$, we consider $\frac{1}{2}$-harmonic maps $u\in H^{1/2}(\partial S,\mathcal{N})$. 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 $\frac{1}{2}$-energies at each boundary component of $\partial S$ (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 Teichmüller space of $S$.

41 pages

References in corpus (1)

• Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

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