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

Abstract

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 $E_2(h)=\int_\mathbb{D}η(h)|h_{\bar w}|^2,dA, \quad\text{equivalently}\quad \int_\mathbb{D}\frac{|μ_f|^2}{1-|μ_f|^2},dA_η,\quad f=h^{-1},$ over all monotone Sobolev maps with boundary values $β$. For Weil–Petersson data, the unique minimiser is a real analytic hyperbolic harmonic diffeomorphism, globally quasiconformal and asymptotically conformal, with an explicit dilatation bound in terms of any admissible barrier's energy. We prove that a circle homeomorphism is Weil–Petersson if and only if it admits a finite-distortion homeomorphic extension satisfying $\int_\mathbb{D}(\mathbb{K}_F-1),dA_η<\infty.$ For every $p>1$, the boundary values of a finite-distortion homeomorphism satisfying $\int_\mathbb{D}|μ_F|^p(1-|μ_F|^2)^{-p_*/2},dA_η<\infty$ lie in Guo's $p$-integrable Teichm"uller space $T^p$. A capacity estimate for boundary moduli for the quasisymmetric ratio yields a derivative-free characterisation of $T^p$, essentially Bishop's theorem when $p=2$. The minimum defines a M"obius-invariant boundary energy, quantitatively comparable to the weighted square norm of the minimiser's Hopf differential. For arbitrary quasisymmetric data, we prove Markovi'c's theorem (the Schoen conjecture) by approximation with our smooth quadratic minimisers. Mori's theorem and the mean-value property of a logarithm verify the averaged-distance hypothesis in Riestenberg–Smillie's interior comparison, giving explicit displacement, energy-density and dilatation bounds in terms of the quasisymmetry constant. The nonquadratic ($p\neq2$) theory is developed in Part~II.