Remarks on Loewner Chains Driven by Finite Variation Functions

TL;DR

This paper investigates Loewner chains driven by finite variation functions, establishing conditions for the existence and continuity of the trace, and exploring when the trace is differentiable, thus linking properties of the driving function to the chain.

Contribution

It provides new conditions under which Loewner chains driven by finite variation functions have continuous traces and differentiability properties.

Findings

Existence of simple trace under certain conditions

Continuity of the trace with respect to the driving function

Conditions for the trace to be continuously differentiable

Abstract

To explore the relation between properties of Loewner chains and properties of their driving functions, we study Loewner chains driven by functions $U$ of finite total variation. Under some appropriate conditions, we show existence of the simple trace $\gamma$ and establish continuity of the map $U$ to $\gamma$ with respect to uniform topology on $\gamma$ and the total variation topology on $U$. In the spirit of work of Wong and Tran-Lind, we also obtain conditions on the driving function that ensures the trace to be continuously differentiable.