The multiple-slit version of Loewner's differential equation and pointwise H\"older continuity of driving functions

TL;DR

This paper extends Loewner's differential equation to multiple slits, exploring how the regularity of driving functions influences the geometry of the generated hulls, including conditions for simplicity and local straightness.

Contribution

It generalizes Lind's result to multiple slits and links local hull geometry to driving function properties in the chordal Loewner framework.

Findings

Generalization of Lind's condition for simple curves to multiple slits

Characterization of hulls locally resembling straight lines at starting points

Conditions relating pointwise H"older continuity of driving functions to hull properties

Abstract

We consider the chordal Loewner differential equation for multiple slits in the upper half-plane and relations between the pointwise H\"older continuity of the driving functions and the generated hulls. The first result generalizes a result of Lind that gives a sufficient condition for driving functions to generate simple curves. The second result translates the property that the hulls locally look like straight lines at their starting points into a condition for the driving functions.