# A construction of trivial Beltrami coefficients

## TL;DR

This paper provides a sufficient condition to determine when a Beltrami coefficient on the unit disk is trivial, utilizing Betker's theorem on L"owner chains, addressing the challenge of explicitly solving the Beltrami equation.

## Contribution

It introduces a new criterion for triviality of Beltrami coefficients based on L"owner chains, advancing understanding of their structure.

## Key findings

- Provides a practical sufficient condition for trivial Beltrami coefficients.
- Utilizes Betker's theorem to connect Beltrami coefficients with L"owner chains.
- Enhances methods for detecting triviality without explicit solutions.

## 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.