Loewner chains and H\"older geometry

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.