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 ⊂ ‫ޒ‬ 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 ⊂ ‫ޒ‬ 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.I thank Amélie Loher and the anonymous referee for careful reading of the manuscript and useful suggestions.