On the Conformal Energy of Quasisymmetric and Quasimöbius Mappings

This article identifies the conformal energy (or mean distortion) of extremal mappings of finite distortion with a given quasisymmetric mapping of the circle as boundary data. The conformal energy of $g_o:\text{IS}\to \text{IS}$ is \begin{equation}\label{energy} \mathcal E(g_o)=-\frac{1}{4π^2}\iint_{$\mathbb{S}$\times $\mathbb{S}$}\log |g_o(ζ)-g_o(η)| : dζd\barη < \infty \end{equation} We give explicit formulae for the conformal energy of circle homeomorphisms directly in terms of their data. As an example, if $g_o:\mathbb{S}\to \mathbb{S}$ is an $\eta$-quasi-Möbius self homeomorphism of the unit circle, then \begin{align*} \mathcal E(g_o) \leq \frac{1}π \int^{π/2}_0 \log η\big[ \cot^2(t/2)\big] ,\cos(t) ; dt \end{align*} This estimate is sharp. Additionally we show how a circle homeomorphism of finite conformal energy can be uniformly approximated on $\text{IS}$ by mappings of strictly smaller energy.