Exact Solutions for Loewner Evolutions

Abstract

In this note, we solve the Loewner equation in the upper half-plane with forcing function ξ(t), for the cases in which ξ(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 ξ(t)=2√kt, the trace is a straight line set at an angle to the real axis. If ξ(t)=2√k(1-t), as pointed out by Marshall and Rohde,⁽¹²⁾ the behavior of the trace as t approaches 1 depends on the coefficient κ. Our calculations give an explicit solution in which for κ<4 the trace spirals into a point in the upper half-plane, while for κ>4 it intersects the real axis. We also show that for κ=9/2 the trace becomes a half-circle. The third case with forcing ξ(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 ξ(t) is a superposition of the values ±1.