Harmonic quasiconformal maps and the hyperbolic metric (Q268046)

The goals of this paper are to place the results of Chen and Fang (2010) for harmonic quasiconformal maps of the unit disk onto convex regions into the general context of Riemannian and conformal metrics and to extend the known results from convex regions to uniformly perfect regions. Let \(\Omega\) be a hyperbolic region in the complex plane \(\mathbb{C}\) and let \(\lambda_\Omega(z)|dz|\) be the hyperbolic metric in \(\Omega\). The region \(\Omega\) is uniformly perfect if there exists \(c(\Omega)>0\) such that \(c(\Omega)/\text{dist}(z,\partial\Omega)\leq\lambda_\Omega(z)\) for all \(z\in\Omega\). For a hyperbolic region \(\Omega\) set \[ N(\Omega)=\sup_{w\in\Omega}\frac{|S_\Omega(w)+(1/2)\Gamma^2_\Omega(w)|}{\lambda^2(w)}. \] It is proved that the constant \(N(\Omega)\) cannot change too dramatically under quasiconformal maps. Let \(\rho(w)|dw|\) be a conformal metric on \(\Omega\), \(f:\tilde\Omega\rightarrow\Omega\) be a locally injective holomorphic function and let \(f^*\) be the pull-back of \(\rho(w)|dw|\) by \(f\). For a \(C^2\)-conformal metric in \(\Omega\) and a harmonic \(k\)-quasiconformal map \(f:\tilde\Omega\rightarrow\Omega\), the curvatures of the upper and lower conformal pull-backs are computed. Furthermore, stronger bounds are obtained in the special case of affine maps \(f(z)=Az+B\bar z\), where \(A,B\in\mathbb{C}\) and \(|A|>|B|\). In the paper, Ahlfors-type inequalities are also considered.