On the contraction of the Teichmüller metrics
A quasiconformal self-mapping \(f\) of the unit disk is extremal in its class if no other quasiconformal self-mapping with the same boundary values on the circumference of the disk has smaller maximal dilatation. The Beltrami coefficient \(\mu (z) = \overline \partial f(z)/ \partial f(z)\) of an extremal quasiconformal mapping sometimes has the form \(\mu = k |\psi |/ \psi\), where \(\psi (z)\) is a holomorphic function in the unit disk with the property that \[ \int \int \bigl |\psi (z) \bigr |dxdy = 1, \quad \text{and} \quad 0 < k < 1. \] This integral and all subsequent integrals in this discussion are taken over the unit disk. Such a Beltrami coefficient is called a Teichmüller Beltrami coefficient with quadratic differential \(\psi\) of finite norm and the associated mapping \(f\) is said to have Teichmüller form. When \(f\) has Teichmüller form, it is uniquely extremal. However, its Beltrami coefficient \(\mu\) may still have degenerating Hamilton sequences. That means there may exist a sequence \(\varphi_n\) of holomorphic functions defined in the unit disk with \[ \int \int \bigl |\varphi_n (z) \bigr |dxdy = 1,\;\varphi_n (z) \text{ converges to zero for each z}, \] \[ \text{and } \lim \int \int \varphi_n (z) \mu (z) dx dy = |\mu |_\infty. \] When \(\mu\) is of the form \(k |\psi |/ \psi\) with \(\psi\) of finite norm, this paper gives a sufficient condition in terms of the geometry of \(\mu\) for \(\mu\) not to possess a degenerating Hamilton sequence. The condition is that there exist a positive number \(r < 1\) such that for almost all \(z\) with \(r < |z |< 1\), \(\arg (\mu (z)/ |\mu (z) |)\) lies between plus and minus \(\theta\), where \(\theta\) is a number larger than \(\pi/2\) and smaller than \(\pi/2 + \arcsin (1/(2C - 1))\). \(C\) is the best number for which the inequality \[ \int \int |\varphi |dxdy \leq C \int \int \bigl |\text{Re} (\varphi) \bigr |dxdy \] for all holomorphic functions \(\varphi\) with \(\text{Im} (\varphi (0)) = 0\) and with \(\int \int |\varphi |dxdy < \infty\). As a consequence, the boundary dilatation of a mapping \(f\) with such a Beltrami coefficient is less that the extremal dilatation of the class of \(f\) and there is proper Teichmüller contraction along the Teichmüller line determined by \(t \mu\), \(- 1/k < t < 1/k\).
scientific article; zbMATH DE number 784740
scientific article