Loewner chains and H\"older geometry

TL;DR

This paper explores the relationship between the analytic properties of Loewner driving functions and the geometric features of the generated domains, with implications for understanding stochastic Loewner evolution (SLE) and H"older domains.

Contribution

It establishes a deterministic link between domain regularity and the modulus of continuity of the driving function, extending understanding of SLE and Loewner chains.

Findings

If the domain is a H"older domain, the driving function has a Brownian-like modulus of continuity.

For simple curves generating H"older domains, the driving function's regularity is characterized.

General geometric criteria are provided for when the driving function is Lip(1/2).

Abstract

The Loewner equation provides a correspondence between continuous real-valued functions $\lambda_t$ and certain increasing families of half-plane hulls $K_t$. In this paper we study the deterministic relationship between specific analytic properties of $\lambda_t$ and geometric properties of $K_t$. Our motivation comes, however, from the stochastic Loewner equation (SLE$_{\kappa}$), where the associated function $\lambda_t$ is a scaled Brownian motion and the corresponding domains $\mathbb{H} \backslash K_t$ are H\"older domains. We prove that if the increasing family $K_t$ is generated by a simple curve and the final domain $\mathbb{H} \backslash K_T$ is a H\"older domain, then the corresponding driving function has a modulus of continuity similar to that of Brownian motion. Informally, this is a converse to the fact that SLE$_{\kappa}$ curves are simple and their complementary domains are H\"older, when $\kappa < 4$. We also study a similar question outside of the simple curve setting, which informally corresponds to the SLE regime $\kappa > 4$. In the process, we establish general geometric criteria that guarantee that $K_t$ has a Lip$(1/2)$ driving function.