Asymptotic conformal welding via Loewner-Kufarev evolution

TL;DR

This paper investigates the asymptotic behavior of conformal mappings generated by the Loewner-Kufarev evolution, establishing variational formulas and connections between different domain types and their conformal radii.

Contribution

It introduces new asymptotic conformal welding results and links driving functions with conformal radii for bounded and unbounded domains.

Findings

Derived variational formulas for asymptotic conformal welding

Established connections between driving functions and conformal radii

Analyzed mappings onto domains close to the unit disk or its exterior

Abstract

The Loewner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk or the exterior of the unit disk. We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Loewner-Kufarev equation and conformal radii of the two domains.