Flowing to free boundary minimal surfaces

TL;DR

This paper introduces a flow that deforms maps from a surface into Euclidean space with boundary conditions into free boundary minimal immersions, combining Plateau-flow and Teichmüller harmonic flow techniques.

Contribution

It develops a new flow that simultaneously deforms the map and the domain metric to produce free boundary minimal immersions for general surfaces.

Findings

The flow converges to a free boundary minimal immersion.

Combining Plateau-flow with Teichmüller harmonic flow addresses non-disc domains.

The method extends minimal surface theory to more general domain surfaces.

Abstract

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ 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 $\Sigma$ 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 $\Sigma\neq D$, 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 Teichm\"uller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial \Sigma\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial \Sigma$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.