Diffeomorphic solutions of Ahlfors-Hopf equations

TL;DR

This paper investigates the boundary value problem for extremal functions of mean distortion, utilizing the Ahlfors-Hopf differential to prove that extremal mappings are local diffeomorphisms, thus resolving longstanding conjectures.

Contribution

It introduces a novel approach using the Ahlfors-Hopf differential to analyze extremal quasiconformal mappings and proves they are local diffeomorphisms inside the domain.

Findings

Extremal mappings are local diffeomorphisms in the unit disk.

The Ahlfors-Hopf differential is holomorphic at extremal points.

The approach resolves existing conjectures in the theory of extremal quasiconformal maps.

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].