Diffeomorphic solutions of Ahlfors-Hopf equations
Gaven Martin, Cong Yao
TL;DR
We address the boundary-value problem for extremal finite-distortion self-maps of the disk using a variational framework built around the Ahlfors–Hopf differential $Φ$ associated with the inverse map $h=f^{-1}$. The approach introduces enlarged energy spaces $F_{A,η}$ and $H_{A,η}$ to guarantee existence and uses inner variations to obtain a holomorphic $Φ$, enabling a demonstration that extremal minimisers are locally diffeomorphic in the disk. Sharp uniqueness results are proved for both disk and automorphic settings, including a unique diffeomorphic minimiser between Riemann surfaces with a holomorphic differential and a precise description of boundary hair-free behaviour. The work further extends to monotone automorphic mappings between quotient surfaces, establishing existence and uniqueness of minimisers in ${\rm Diff}(\Sigma_1,\Sigma_2)$ under general weighting, with $Φ$ belonging to $L^1$ in the appropriate regime.
Abstract
Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichmüller 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 \[ Φ=\mathcal{A}(\mathbb{K}(w,h))h_w\,\overline{h_{\overline{w}}}\, η(h), \] 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 $Φ$ 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].