Plateau flow or the heat flow for half-harmonic maps

TL;DR

This paper develops a classical approach to the heat flow for half-harmonic maps from the circle to a target manifold, establishing results analogous to harmonic map heat flow and linking to minimal surface Plateau problems.

Contribution

It introduces a new classical method for analyzing the heat flow of half-harmonic maps, extending previous work and connecting to minimal surface theory.

Findings

Established existence results for finite-energy data

Connected half-harmonic map heat flow to Plateau problem

Extended classical harmonic map results to half-harmonic setting

Abstract

Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $\Gamma$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.