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The dependence of capacities on moving branch points. (English) Zbl 1133.31002

Let \(A\) and \(B\) be disjoint non-degenerate continua in \(\mathbb{C}\) with connected complements; a pasting arc for \(A\) and \(B\) is defined to be a simple arc \(\gamma\) in the sphere \(\widehat{\mathbb{C}}\) that is disjoint from \(A\cup B\). Then, by forming a covering Riemann surface \(\widehat{\mathbb{C}}_\gamma\) over \(\widehat{\mathbb{C}}\) by pasting two copies of \(\widehat{\mathbb{C}}\setminus\gamma\) crosswise across the arc \(\gamma\), we may view \(A\) and \(B\) as embedded in the two different sheets \(\widehat{\mathbb{C}}\setminus\gamma\) of \(\widehat{\mathbb{C}}_\gamma\) and denote by \(\text{cap}(A, \widehat{\mathbb{C}}_\gamma\setminus B)\) the variational 2-capacity of the set \(A\) in \(\widehat{\mathbb{C}}_\gamma\) with respect to the open subset \(\widehat{\mathbb{C}}_\gamma\setminus B\) containing \(A\).
The author obtains a variational formula for this capacity, from which he shows that the capacity changes smoothly as one branch point moves. The material is related to a series of the author’s papers; for instance, see M. Nakai and S. Segawa [Complex Var., Theory Appl. 49, No. 4, 229–240 (2004; Zbl 1060.30053)].

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C85 Capacity and harmonic measure in the complex plane
30F15 Harmonic functions on Riemann surfaces

Citations:

Zbl 1060.30053
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References:

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