Steffen Rohde

Basic properties of SLE

SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.

The Loewner differential equation and slit mappings

In his study of extremal problems for univalent functions, K. Lowner [11] (who later changed his name into C. Loewner) introduced the differential equation named after him. It was a key ingredient in the proof of the Bieberbach conjecture by de Branges [2]. It was used by L. Carleson and N. Makarov in their investigation of a process similar to DLA [3]. Recently, O. Schramm [20] found a description of the scaling limits of some stochastic processes in terms of the Loewner equation (assuming the validity of some conjectures such as existence of the limits). This led him to the definition of a new stochastic process, the “Stochastic Loewner Evolution” (SLE). The SLE has been further explored in the work of Lawler, Schramm and Werner (see [14] and the references therein) and led to a proof of Mandelbrot’s conjecture about the Hausdorff dimension of the Brownian frontier. The SLE has also played a crucial role in S. Smirnov’s [21] recent work on percolation. Somewhat surprisingly, the geometry of the solutions to the Loewner equation is not very well understood. This paper addresses the regularity of solutions in the deterministic setting. Let Ωt, t0 ≤ t ≤ t1, be a continuously increasing sequence of simply connected planar domains, and let z0 ∈ ⋂ t Ωt. Then there are conformal maps ft : D → Ωt of the unit disc with ft(0) = z0 and f ′ t(0) > 0. The continuity of the domain sequence can be expressed by saying that the map t → ft is continuous in the topology of locally uniform convergence. Since f ′ t(0) is increasing in t, we may assume (by reparametrizing Ωt if necessary) that f ′ t(0) = e t for t0 ≤ t ≤ t1. The family ft, t0 ≤ t ≤ t1, is called a (normalized) Loewner chain. The Loewner differential equation

Convergence of a Variant of the Zipper Algorithm for Conformal Mapping

In the early 1980s an elementary algorithm for computing conformal maps was discovered by R. Kuhnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points

Collisions and spirals of Loewner traces

z_0,\dots,z_n

Rigidity of holomorphic Collet-Eckmann repellers

in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve

Quasicircles modulo bilipschitz maps

\gamma

The Gehring–Hayman inequality for quasihyperbolic geodesics

with

On the Riemann surface type of random planar maps

z_0,\dots,z_n \in \gamma

Oded Schramm: From circle packing to SLE.

. We prove convergence for Jordan regions in the sense of uniformly close boundaries and give corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a

Schramm–Loewner Equations Driven by Symmetric Stable Processes

C^1