The Loewner driving function of trajectory arcs of quadratic differentials

TL;DR

This paper derives a first order differential equation for the Loewner driving function when the domain is slit by a trajectory arc of quadratic differentials, enabling numerical calculation and extending to various Loewner variants.

Contribution

It introduces a novel differential equation for the Loewner driving function in domains bounded by quadratic differential trajectory arcs, with applications to lattice paths and multiple slit configurations.

Findings

Derived a differential equation for the Loewner driving function in specific domains.

Demonstrated numerical computation of the driving function using the new formula.

Extended the results to multiple slits and radial Loewner equations.

Abstract

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The method also works for other versions of the Loewner differential equation. The proof of our formula uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential that is of interest in its own right.