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2007 The dependence of capacities on moving branch points
Mitsuru Nakai
Nagoya Math. J. 186: 1-27 (2007).


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.


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Mitsuru Nakai. "The dependence of capacities on moving branch points." Nagoya Math. J. 186 1 - 27, 2007.


Published: 2007
First available in Project Euclid: 22 June 2007

zbMATH: 1133.31002
MathSciNet: MR2334363

Primary: 31A15
Secondary: 30C85 , 30F15

Keywords: branch point , capacity , covering surface , directional derivative , Dirichlet integral , Dirichlet principle , harmonic measure , standard local parameter , two sheeted sphere

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal


Vol.186 • 2007
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