Radial extension of a bi-Lipschitz parametrization of a starlike Jordan curve

TL;DR

This paper investigates the radial extension of bi-Lipschitz parameterizations of starlike Jordan curves, proving conditions under which the extension remains bi-Lipschitz and quasiconformal, with explicit Lipschitz constant relations.

Contribution

It establishes that the radial extension of a bi-Lipschitz parameterization of a starlike Jordan curve is also bi-Lipschitz and quasiconformal, providing explicit Lipschitz constant comparisons.

Findings

Radial extension preserves bi-Lipschitz property for starlike Jordan curves.

If the curve is a circle centered at the origin, Lipschitz constants are equal.

For non-circular curves, the Lipschitz constant of the extension exceeds that of the boundary parameterization.

Abstract

In this paper we discus the radial extension $w$ of a bi-Lipschitz parameterization $F(e^{it})=f(t)$ of a starlike Jordan curve $\gamma$ w.r. to 0. We show that, if parameterization is bi-Lipschitz, then the extension is bi-Lipschitz and consequently quasiconformal. If $\gamma$ is the unit circle, then $\mathrm{Lip}(f)=\mathrm{Lip}(F)=\mathrm{Lip}(w)=K_w$. If $\gamma$ is not a circle centered at origin, and $F$ is a polar parametrization of $\gamma$, then we show that $\mathrm{Lip}(f)=\mathrm{Lip}(F)<\mathrm{Lip}(w)$.