Trace for the Loewner Equation with Singular Forcing

TL;DR

This paper analyzes the structure of the trace in the Loewner equation with a singular forcing proportional to (-t)^beta, revealing a simple curve touching the real axis twice, using asymptotic analysis and numerical support.

Contribution

It provides a detailed asymptotic description of the trace for singular forcing functions in the Loewner equation, a case not extensively studied before.

Findings

The trace is a simple curve touching the real axis twice.

Asymptotic shape of the trace near the singularity is characterized.

Numerical calculations support the theoretical asymptotic results.

Abstract

The Loewner equation describes the time development of an analytic map into the upper half of the complex plane in the presence of a "forcing", a defined singularity moving around the real axis. The applications of this equation use the trace, the locus of singularities in the upper half plane. This note discusses the structure of the trace for the case in which the forcing function, xi(t), is proportional to (-t)^beta with beta in the interval (0, 1/2). In this case, the trace is a simple curve, gamma(t), which touches the real axis twice. It is computed by using matched asymptotic analysis to compute the trajectory of the Loewner evolution in the neighborhood of the singularity, and then assuming a smooth mapping of these trajectories away from the singularity. Near the t=0 singularity, the trace has a shape given by [ Re(gamma(t)-gamma(0)) ]^(1-beta) ~ [ beta*Im(gamma(t)) ]^beta ~ O(xi(t))^(1-beta). A numerical calculation of the trace provides support for the asymptotic theory.