Flowing to free boundary minimal surfaces

Abstract

We introduce a flow that is designed to flow maps $u:\hat{I}\pounds \to {\mathbb{R}}^n$ which map the boundary of a general domain surface $\hat{I}\pounds$ into a given (not necessarily connected) submanifold $N\hookrightarrow {\mathbb{R}}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $\hat{I}\pounds$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $\hat{I}\pounds \ne D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0:\partial \hat{I}\pounds \to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial \hat{I}\pounds$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.