Singular solutions to the Loewner equation

Abstract

We consider the Löwner differential equation generating univalent self-maps of the unit disk (or of the upper half-plane). If the solution to this equation represents a one-slit map, then the driving term is a continuous function. The reverse statement is not true in general as a famous Kufarev's example shows. We address the following main problem: to find a criterium for the Löwner equation to generate one-slit solutions. New examples of non-slit solutions to the Löwner equation are presented. Properties of singular slit solutions are revealed.

14 pages, figures are added