Geodesic Loewner paths with varying boundary conditions

TL;DR

This paper formulates and solves Loewner equations with non-constant boundary conditions to model geodesic growth paths of slits in the upper half-plane, revealing diverse growth behaviors influenced by boundary conditions.

Contribution

It introduces a novel formulation of Loewner equations with varying boundary conditions and analyzes the resulting geodesic slit paths in the context of Laplacian growth.

Findings

Paths can be bounded, unbounded, or terminate finitely depending on boundary conditions.

Symmetric paths tend to bifurcate at an angle of π/5 asymptotically.

Boundary conditions significantly influence slit growth behavior.

Abstract

Equations of the Loewner class subject to non-constant boundary conditions along the real axis, are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped `mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of $\pi/5$.