Plateau flow or the heat flow for half-harmonic maps

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 $N\subset {ℝ}^n$, 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 $\Gamma \subset {ℝ}^n$, 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.