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Using the interpretation of the half-Laplacian on $S^1$ as theDirichlet-to-Neumann operator for the Laplace equation on the ball $B$, wedevise 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 forarbitrary finite-energy data we obtain a result fully analogous to the author's1985 results for the harmonic map heat flow of surfaces and in similargenerality. When $N$ is a smoothly embedded, oriented closed curve $\Gamma$ thehalf-harmonic map heat flow may be viewed as an alternative gradient flow for avariant of the Plateau problem of disc-type minimal surfaces.

Plateau flow or the heat flow for half-harmonic maps