2017/04/26 by Sugawa, Toshiyuki
#30F60 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C62 #Secondary 30C55
paper · doi:10.48550/arxiv.1704.07951
A measurable function μ on the unit disk \mathbbD of the complex plane with ‖μ‖_∞<1 is sometimes called a Beltrami coefficient. We say that μ is trivial if it is the complex dilatation f z/fz of a quasiconformal automorphism f of \mathbbD 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 Löwner chains.