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A Loewner Equation for Infinitely Many Slits

A Loewner Equation for Infinitely Many Slits It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union $$\Gamma $$ Γ of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of $$\Gamma $$ Γ , our approach allows us to derive a Loewner equation for infinitely many slits as well. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Loewner Equation for Infinitely Many Slits

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Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-016-0179-6
Publisher site
See Article on Publisher Site

Abstract

It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union $$\Gamma $$ Γ of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of $$\Gamma $$ Γ , our approach allows us to derive a Loewner equation for infinitely many slits as well.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Oct 3, 2016

References