Chordal Loewner Equation

TL;DR

This survey provides a straightforward proof that conformal maps of the upper-half plane evolving over time satisfy the chordal Loewner differential equation, emphasizing elementary geometric and topological methods.

Contribution

It offers a complete, elementary proof of the chordal Loewner equation for hydrodynamically normalized conformal maps, avoiding advanced techniques.

Findings

Proof based solely on basic geometric function theory

Clarifies the relationship between conformal maps and Loewner equation

Simplifies understanding of the chordal Loewner evolution

Abstract

The aim of this survey paper is to present a complete direct proof of the well celebrated cornerstone result in Loewner Theory, originally due to Kufarev et al [Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 200 (1968) 142-164. MR0257336 (41 #1987)], stating that the family of the hydrodynamically normalized conformal self-maps of the upper-half plane onto the complement of a gradually erased slit satisfies, under a suitable parametrization, the chordal Loewner differential equation. The proof is based solely on basic theorems of Geometric Function Theory combined with some elementary topological facts and does not require any advanced technique.