[Paper Review] Loewner chains on the universal covering space of a Riemann surface

Jonathan Tsai|ArXiv.org|Dec 19, 2008

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper extends the classical Loewner differential equation to Riemann surfaces with boundary by lifting curves to the universal covering space (the upper half-plane), deriving a time-evolving conformal map family that satisfies a generalized Loewner equation. The key contribution is a differential equation governing the evolution of conformal maps on the universal cover, enabling a foundation for stochastic Loewner evolution (SLE) on multiply-connected domains and Riemann surfaces.

ABSTRACT

Let R be a hyperbolic Riemann surface with boundary $\partial R$ and suppose that $γ:[0,T] o R\cup\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\setminus γ(0,t]$ 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 $π:\mathbb{H} o R$, we can define a family of simply-connected domains $D_{t}=π^{-1}(R_{t})$. 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.

Motivation & Objective

• To generalize the chordal Loewner differential equation to Riemann surfaces with boundary, particularly hyperbolic Riemann surfaces.
• To address the lack of a natural normalization for Loewner chains on multiply-connected domains.
• To provide a framework for defining stochastic Loewner evolution (SLE) on Riemann surfaces by lifting curves to the universal cover.
• To establish a differential equation for conformal maps on the universal cover that evolve with time as a curve grows on the base surface.
• To enable the construction of SLE on Riemann surfaces by characterizing the driving function and group structure under conformal evolution.

Proposed method

• Lift a simple curve γ on a hyperbolic Riemann surface R with boundary to the universal cover, which is the upper half-plane H, via the covering map π: H → R.
• Define the family of simply-connected domains Dt = π⁻¹(Rt), where Rt = R ∖ γ(0,t], so that Dt ⊂ H.
• Construct a family of conformal maps ft: H → Dt such that ft is differentiable a.e. in t ∈ (0,T), forming a Loewner chain.
• Derive a differential equation for ft(z) = f(z,t) that describes its time evolution, generalizing the classical chordal Loewner equation.
• Use the Fuchsian group Γ associated with R to define the group of deck transformations and track the evolution of the group Γt = ft⁻¹ ∘ Γ ∘ ft.
• Establish the consistency of the Loewner chain by showing that the induced Fuchsian group Γt and conformal map gt are independent of the choice of initial points, ensuring well-defined evolution.

Experimental results

Research questions

• RQ1 How can the classical Loewner differential equation be generalized to Riemann surfaces with boundary, particularly in the multiply-connected case?
• RQ2 What conditions ensure the existence and differentiability of a Loewner chain on the universal cover of a Riemann surface?
• RQ3 How does the Fuchsian group structure evolve under the conformal evolution of the covering space?
• RQ4 Can stochastic Loewner evolution (SLE) be consistently defined on Riemann surfaces by extending the driving function to include drift and diffusion terms?
• RQ5 What normalization and parameterization schemes are suitable for Loewner chains on Riemann surfaces, given the absence of a natural one in the multiply-connected setting?

Key findings

• The paper derives a differential equation for the time evolution of the conformal map ft(z) on the universal cover H, generalizing the chordal Loewner equation to Riemann surfaces.
• The evolution of ft is governed by a differential equation involving the inverse image of the curve's endpoint under ft, analogous to the classical Loewner equation.
• The Fuchsian group Γt associated with the punctured surface R ∖ γ(0,t] evolves via conjugation by ft, ensuring the conformal equivalence R ∖ γ(0,t] ≅ H / Γt.
• The Loewner chain is independent of the choice of initial points in R ∖ Λ, ensuring consistency of the construction across different parameterizations.
• The framework supports the definition of SLE on Riemann surfaces by allowing the driving function to include a drift term h(t)dt and a Brownian motion term √κ dBt, generalizing chordal SLE.
• The construction is valid for finite Riemann surfaces (finitely generated Fuchsian groups), and extends to more general cases via approximation.