Harmonic extension of Weil-Petersson circle homeomorphisms

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.