Singular solutions to the Loewner equation

Details Singular solutions to the Loewner equation Singular solutions to the Loewner equation

2008-07-03

arxiv.org/abs/0807.0519v2

Mathematics

Complex Variables

14 pages, figures are added

Scientific paper

We consider the L\"owner 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 L\"owner equation to generate one-slit solutions. New examples of non-slit solutions to the L\"owner equation are presented. Properties of singular slit solutions are revealed.

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