On driving functions generating quasislits in the chordal Loewner-Kufarev equation

TL;DR

This paper constructs specific driving functions for the chordal Loewner-Kufarev equation that generate quasislits, demonstrating how the boundary behavior of the driving function influences the geometric shape of the generated slit.

Contribution

It proves the existence of driving functions with prescribed boundary oscillation that produce quasislits in the Loewner-Kufarev evolution, linking boundary behavior to geometric outcomes.

Findings

Existence of driving functions with prescribed boundary oscillation

Generation of quasislits via specific driving functions

Connection between boundary behavior and slit geometry

Abstract

We prove that for every $C>0$ there exists a driving function $U:[0,1]\to\mathbb{R}$ such that the corresponding chordal Loewner-Kufarev equation generates a quasislit and $ \limsup_{h\downarrow0}\frac{|U(1)-U(1-h)|}{\sqrt{h}}=C. $