Integrable distortion I: Weil--Petersson circle homeomorphisms and the Schoen conjecture

A circle homeomorphism $β:\mathbb{S}\to\mathbb{S}$ is Weil--Petersson if it has a quasiconformal disk extension whose Beltrami coefficient is square integrable for hyperbolic area. We minimise the hyperbolic anti-holomorphic energy \[ E_2(h)=\int_\mathbb{D}η(h)|h_{\bar w}|^2,dA, \quad\text{equivalen