PLATEAU FLOW OR THE HEAT FLOW FOR HALF-HARMONIC MAPS

Abstract

Using the interpretation of the half-Laplacian on S1 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 S1 to a closed target manifold N ⊂ Rn, 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 0 ⊂ Rn, 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.