Singular and tangent slit solutions to the Loewner equation

TL;DR

This paper investigates the Loewner differential equation, characterizing the generation of slit domains via specific driving functions, and identifies conditions for quasisymmetric slit formation with a focus on tangent and singular solutions.

Contribution

It provides a detailed analysis of tangent and singular slit solutions in the Loewner equation, including the regularity of driving functions and the critical norm for quasisymmetric slits.

Findings

Circular slits tangent to the real axis are generated by Hölder continuous driving terms with exponent 1/3.

Singular solutions to the Loewner equation are characterized.

The critical norm value for driving terms producing quasisymmetric slits is determined.

Abstract

We consider the Loewner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by H\"older continuous driving terms with exponent 1/3 in the Loewner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.