A Sharp Condition for the Loewner Equation to Generate Slits

TL;DR

This paper establishes the maximum constant for the H"older continuous driving term in the Loewner equation to generate slits, refining the understanding of the equation's geometric behavior.

Contribution

The paper proves that the maximal value of the H"older norm for the driving term in the Loewner equation to produce slits is exactly 4, providing a sharp condition.

Findings

Maximum H"older norm for slit generation is 4

Sharp condition improves previous bounds

Clarifies the boundary between slit and non-slit generation

Abstract

D. Marshall and S. Rohde have recently shown that there exists $C_0 >0$ so that the Loewner equation generates slits whenever the driving term is H\"older continuous with exponent 1/2 and norm less than $C_0$. In this paper, we show that the maximal value for $C_0$ is 4.