Basins of attraction in Loewner equations

TL;DR

This paper investigates the dynamics of Loewner PDEs with specific conditions on the driving term, establishing existence and geometric properties of solutions via discretization and basin of attraction analysis.

Contribution

It introduces conditions under which Loewner PDE solutions are univalent and describes the biholomorphic nature of their union of images.

Findings

Solutions exist under certain bunching conditions.

Union of solution images is biholomorphic to complex space for r<2.

Method involves discretizing time and analyzing basins of attraction.

Abstract

We prove that any Loewner PDE whose driving term h(z,t) vanishes at the origin, and satisfies the bunching condition r m(Dh(0,t))\geq k(Dh(0,t)) for some r\in R^+, admits a solution given by univalent mappings (f_t: B^q\to C^q). This is done by discretizing time and considering the abstract basin of attraction. If r<2, then the union of the images f_t(\B^q) of a such solution is biholomorphic to C^q.