Drivers, hitting times, and weldings in Loewner's equation

TL;DR

This paper investigates the relationships between driving functions, hitting times, and conformal weldings in Loewner's equation, revealing new continuity properties and approximation results for curves with finite Loewner energy.

Contribution

It introduces new continuity results linking driver convergence to hitting time and welding convergence, and demonstrates approximation of finite energy curves by energy minimizers.

Findings

Uniform driver convergence implies hitting time and welding convergence.

Welding convergence does not imply hitting time or driver convergence.

Hitting time convergence implies driver convergence for constant drivers.

Abstract

In addition to conformal weldings $\varphi$, simple curves $\gamma$ growing in the upper half plane generate driving functions $\xi$ and hitting times $\tau$ through Loewner's differential equation. While the Loewner transform $\gamma \mapsto \xi$ and its inverse $\xi \mapsto \gamma$ have been carefully examined, less attention has been paid to the maps $\xi \mapsto \tau \mapsto \varphi$. We study their continuity properties and show that uniform driver convergence implies uniform hitting time convergence and uniform welding convergence, even when the corresponding curves do not converge. Welding convergence implies neither hitting time nor driver convergence, while hitting time convergence implies driver convergence in (at least) the case of constant drivers. As an application, we show that a curve $\gamma$ of finite Loewner energy can be well approximated by an energy minimizer that matches $\gamma$'s welding on a sufficiently-fine mesh.