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The Dependence of Capacities on moving Branch points

Published online by Cambridge University Press:  11 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
Department of Mathematics Nagoya Institute of Technology Gokiso, Showa Nagoya, 466-8555Japan
*
52 Eguchi, Hinaga, Chita, 478-0041, Japan, nakai@daido-it.ac.jp
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Abstract

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We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

Keywords

31A1530C8530F15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 186 , 2007 , pp. 1 - 27
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

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