Non-slit and singular solutions to the Löwner equation

We consider the Löwner differential equation in ordinary derivatives 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, a...

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Bibliographic Details
Published inBulletin des sciences mathématiques Vol. 136; no. 3; pp. 328 - 341
Main Authors Ivanov, Georgy, Prokhorov, Dmitri, Vasilʼev, Alexander
Format Journal Article
LanguageEnglish
Published Elsevier Masson SAS 01.04.2012
Subjects
Löwner equation
Slit map
Univalent function
secondary
Slit map
Univalent function
Löwner equation
primary
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Summary:We consider the Löwner differential equation in ordinary derivatives 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 Kufarevʼs example shows. Lind, Marshall and Rohde found a sufficient condition for the driving term in the Löwner equation which guarantees a slit solution. The 1/2 Lipschitz norm of this term must be less than 4. We construct a family of non-slit solutions to the Löwner equation whose driving term is of 1/2 Lipschitz norm which admits the whole spectrum of values [4,∞). Then we turn to the properties of singular slit solutions in the half-plane. In particular, we prove that an analytic orthogonal slit is 1/2 Lipschitz with the vanishing norm.
ISSN:0007-4497
1952-4773
DOI:10.1016/j.bulsci.2011.09.002