Harmonic extension of Weil-Petersson circle homeomorphisms

Abstract

In this paper, we study Weil--Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ is Weil--Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\mathbb{D}$ has square-integrable Beltrami differential. Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil--Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extension minimizes this energy.

15 pages