Loewner chains and H\"older geometry

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 $\mathbb{H}\mathrm{\backslash }{K}_t$ are H"older domains. We prove that if the increasing family $K_t$ is generated by a simple curve and the final domain $\mathbb{H}\mathrm{\backslash }{K}_T$ 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 $\kappa > 4$\mathrm{.}Intheprocess,weestablishgeneralgeometriccriteriathatguaranteethat$K_t$hasaLip$(1/2)$ driving function.