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On Conformal Mappings Onto Incised Radial Slit Disks by Kôtaro Oikawa; Nobuyuki Suita is a Mathematics article available to read on EtoBox.

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A plane domain cannot always be mapped conformally onto a radial slit disk. But it was shown by Strebel 15] and Reich [3] that it can be mapped onto an incised radial slit disk. We are interested in the problem: under what circumstances do these incisions occur. In the present paper we shall show that the occurrence depends only on a property of a neighborhood of the " boundary element " corresponding to the incision. The concept of such a boundary element was introduced recently by the second author [8,9]. We shall define it here in a somewhat different manner. ## Statement of results. 1. Let Ω be a proper subdomain on the Riemann sphere, ζeβ, and γ be a boundary component of Ω. For the sake of simplicity assume ζ^oo. Let be the family of the functions / with the following properties: / is regular and univalent on £,/(ζ)=0, /'(ζ)=l, and/fr) is the other boundary component of f(Q). We introduce the quantity R(γ}=R(Ω, ζ, γ) by log Λ( 7)=Iίm (log e+2πλ(Γ\*)). ## e->0 Here ε>0 satisfies {z\\z-ζ\^ε}c:Ω, λ stands for extremal length, and Γ\* is the family of the locally rectifiable open arcs in Ω-{z\\z-ζ\^ε} joining r and the circle \z-ζ|= e . By an open arc in a domain we mean a contin

Who reads On Conformal Mappings Onto Incised Radial Slit Disks?

Author

Kôtaro Oikawa; Nobuyuki Suita

Publisher

Japan Science and Technology Information Aggregator, Electronic; Tokyo Institute of Technology, Department of Mathematics; Tokyo Institute of Technology (ISSN 0386-5991)

Published

1970

Language

EN

Field

Mathematics (Physical Sciences)