A construction of trivial Beltrami coefficients

Abstract

A measurable function $\mu$ on the unit disk $\mathbb{D}$ of the complex plane with $\|\mu\|_\infty<1$ is sometimes called a Beltrami coefficient. We say that $\mu$ is trivial if it is the complex dilatation $f_{\bar z}/f_z$ of a quasiconformal automorphism $f$ of $\mathbb{D}$ satisfying the trivial boundary condition $f(z)=z,~|z|=1.$ Since it is not easy to solve the Beltrami equation explicitly, to detect triviality of a given Beltrami coefficient is a hard problem, in general. In the present article, we offer a sufficient condition for a Beltrami coefficient to be trivial. Our proof is based on Betker's theorem on L\"owner chains.