Diffeomorphic solutions of Ahlfors-Hopf equations

Abstract

Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichm\"uller spaces interpolating between the classical examples of extremal quasiconformal mappings, and the more recent approach through harmonic mappings (of extreme Dirichlet energy). In this paper we focus on the Alhfors-Hopf differential \[ \Phi=\mathcal{A}(\mathbb{K}(w,h))h_w\,\overline{h_{\overline{w}}}\, \eta(h), \] where $h=f^{-1}$ is the pseudo-inverse of an extremal mapping $f$ for the problem \[ \inf_{f:\mathbb{D}\to\mathbb{D}}\int_\mathbb{D} \mathcal{A}(\mathbb{K}(z,f)) \; dz, \quad\quad \mathbb{K}(z,f) = \frac{|f_z|^2+|f_{\overline{z}}|^2}{|f_z|^2-|f_{\overline{z}}|^2}. \] where the infimum is taken over those homeomorphisms of finite distortion $f:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ with $f|\mathbb{S}=f_0$, typically a quasisymmetric barrier function. The inner-variational equations, an analogue of the Euler-Lagrange equations, show $\Phi$ is holomorphic at an extremal. Exploiting this Ahlfors-Hopf differential, we prove that an extreme point $f$ is a local diffeomorphism in $\mathbb{D}$, resolving some conjectures in [16].