Loewner Chains on the Universal Covering Space of a Riemann Surface

Let R be a hyperbolic Riemann surface with boundary ∂R and suppose that γ : [0, T ] → R ∪ ∂R is a simple curve with γ(0, T ] ⊂ R and γ(0) ∈ ∂R. By lifting Rt = R \ γ(0, t] to the universal covering space of R (which we assume is the upper half-plane H = {z ∈ C : Im[z] > 0}) via the covering map π : H → R, we can define a family of simply-connected domains Dt = π(Rt) ⊂ H. For each t ∈ [0, T ], suppose that ft is a conformal map of H onto Dt such that f(z, t) = ft(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 to multiply-connected domains and Riemann surfaces.