Loewner chains on the universal covering space of a Riemann surface

Abstract

Let R be a hyperbolic Riemann surface with boundary $\partial R$ and suppose that $\gamma:[0,T]\to R\cup\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\setminus \gamma(0,t]$ to the universal covering space of R (which we assume is the upper half-plane $\mathbb{H}=\{z\in\mathbb{C}:Im[z]>0\}$) via the covering map $\pi:\mathbb{H}\to R$, we can define a family of simply-connected domains $D_{t}=\pi^{-1}(R_{t})$. For each $t\in[0,T]$, suppose that f_{t} is a conformal map of \mathbb{H} onto D_{t} such that f(z,t)=f_{t}(z) is differentiable almost everywhere in (0,T) with respect to t. In this paper, we will derive a differential equation that describes how f(z,t) evolves in time t. This should be viewed as an extension of the Loewner differential equation to curves on Riemann surfaces with boundary. The motivation of this paper is the desire to extend Schramm's stochastic Loewner evolution (SLE) to multiply-connected domains and Riemann surfaces.