Exact Solutions for Loewner Evolutions

TL;DR

This paper provides explicit solutions to the Loewner equation in the upper half-plane for specific power-law forcing functions, revealing diverse trace behaviors including spirals, intersections, and circular shapes.

Contribution

It offers exact solutions for the Loewner equation with power-law and superposed forcing functions, expanding understanding of trace geometries in complex analysis.

Findings

Trace is perpendicular to the real axis for xi(t)=constant.

Trace can be a straight line, spiral, or circle depending on xi(t).

Behavior of the trace varies with the parameter kappa, including spirals and intersections.

Abstract

In this note, we solve the Loewner equation in the upper half-plane with forcing function xi(t), for the cases in which xi(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If xi(t)=2*(kappa*t)^1/2, the trace is a straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2, the behavior of the trace as t approaches 1 depends on the coefficient kappa. Our calculations give an explicit solution in which for kappa<4 the trace spirals into a point in the upper half-plane, while for kappa>4 it intersects the real axis. We also show that for kappa=9/2 the trace becomes a half-circle. The third case with forcing xi(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which xi(t) is a superposition of the values +1 and -1.