Harmonic maps from post-critically finite fractals to the circle

Abstract: Every continuous map between two compact Riemannian manifolds is homotopic to a harmonic map (HM). We show that a similar situation holds for continuous maps between a post-critically finite (p.c.f.) fractal and a circle. Specifically, we provide a geometric proof of Strichartz's theorem stating that for a given degree and appropriate boundary conditions there is a unique HM from the Sierpinski Gasket (SG) to a circle. Furthermore, we extend this result to HMs on p.c.f. fractals. Our method uses covering spaces for the SG, which are constructed separately for HMs of a given degree, thus capturing the topology intrinsic to each homotopy class. After lifting continuous functions on the SG with values in the unit circle to continuous real-valued functions on the covering space, we use the harmonic extension algorithm to obtain a harmonic function on the covering space. The desired HM is obtained by restricting the domain of the resultant harmonic function to the fundamental domain and projecting the range to the circle. We show that with suitable modifications the method applies to p.c.f. fractals, a large class of self-similar domains. We illustrate our method of constructing the HMs using numerical examples of HMs from the SG to the circle and discuss the construction of the covering spaces for several representative p.c.f. fractals, including the 3-level SG, the hexagasket, and the pentagasket.