A construction of trivial Beltrami coefficients

Sugawa, Toshiyuki

doi:10.1090/proc/13965

A construction of trivial Beltrami coefficients
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by Toshiyuki Sugawa PDF

Proc. Amer. Math. Soc. 147 (2019), 629-635 Request permission

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 Löwner chains.

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