Evolution Families and the Loewner Equation I: The Unit Disc

TL;DR

This paper introduces a unified Loewner differential equation framework for the unit disc, linking radial and chordal cases, and explores the properties of evolution families and their generating vector fields.

Contribution

It presents a new generalized Loewner equation that unifies previous formulations and establishes a correspondence with evolution families in the unit disc.

Findings

Establishes a one-to-one correspondence between evolution families and solutions to the new Loewner equations.

Provides a Berkson-Porta type formula for non-autonomous holomorphic vector fields.

Analyzes geometric and dynamical properties of evolution families.

Abstract

In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson-Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.