Spacefilling Curves and Phases of the Loewner Equation

TL;DR

This paper establishes a deterministic phase transition for Loewner traces from simple to space-filling curves at a specific Lip(1/2) norm threshold, extending understanding of geometric properties of these curves.

Contribution

It identifies a critical Lip(1/2) norm value for space-filling Loewner traces and provides geometric criteria for driving functions, including examples like space-filling curves and fractals.

Findings

Existence of a constant C>4 for space-filling Loewner traces with Lip(1/2) norm

Geometric criteria for Lip(1/2) driven traces

Examples include Hilbert space-filling curve and Sierpinski gasket

Abstract

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C>4 such that a Loewner driving term whose trace is space filling has Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpinski gasket fall into this class.