The Loewner energy of loops and regularity of driving functions

TL;DR

This paper establishes the regularity properties of Loewner driving functions for smooth curves and introduces the Loewner energy of loops, demonstrating its independence from the basepoint, thus advancing understanding of conformal geometry and Loewner theory.

Contribution

The paper proves optimal regularity results for Loewner driving functions of smooth curves and introduces the Loewner energy of loops, showing its basepoint independence.

Findings

Loewner driving functions of $C^{1,eta}$ curves are in $C^{1,eta-1/2}$ for $eta>1/2$

Loewner driving functions are in $C^{0,eta+1/2}$ for $0 extlesseta extless=1/2

Loewner energy of loops is independent of the basepoint

Abstract

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,\beta}$ curve (differentiable parametrization with $\beta$-H\"older continuous derivative) is in the class $C^{1,\beta-1/2}$ if $1/2<\beta\leq 1$, and in the class $C^{0,\beta + 1/2}$ if $0 \leq \beta \leq 1/2$. This is the converse of a result of Carto Wong and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint.