Regularity of Loewner Curves

TL;DR

This paper establishes a direct relationship between the regularity of the driving function in the Loewner equation and the resulting curve's smoothness, showing higher regularity in the driving function yields smoother curves.

Contribution

It proves that if the driving function is sufficiently smooth, then the generated Loewner curve inherits increased regularity, extending previous results and providing a converse to known theorems.

Findings

If $eta>2$, the Loewner curve is in $C^{eta + 1/2}$.

Analytic driving functions produce analytic Loewner curves.

Extends and complements prior regularity results by Earle, Epstein, and Wong.

Abstract

The Loewner equation encrypts a growing simple curve in the plane into a real-valued driving function. We show that if the driving function $\lambda$ is in $C^{\beta}$ with $\beta>2$ (or real analytic) then the Loewner curve is in $C^{\beta + \frac{1}{2}}$ (respectively analytic). This is a converse to a result by Earle and Epstein and extends a result of Wong.