Exponential driving function for the L\"owner equation

TL;DR

This paper investigates the solutions of the chordal Löwner differential equation driven by a cube root function, analyzing their properties, monotonicity, and geometric mappings of slit domains onto the upper half-plane.

Contribution

It introduces a detailed analysis of solutions driven by a cube root function, including their series representations and geometric properties of the resulting slit domains.

Findings

Solutions map slit domains onto the upper half-plane

Slit is a $C^1$-curve

Provides asymptotic estimate for harmonic measure ratio

Abstract

We consider the chordal L\"owner differential equation with the model driving function $\root3\of t$. Holomorphic and singular solutions are represented by their series. It is shown that a disposition of values of different singular and branching solutions is monotonic, and solutions to the L\"owner equation map slit domains onto the upper half-plane. The slit is a $C^1$-curve. We give an asymptotic estimate for the ratio of harmonic measures of the two slit sides.