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)].


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


Zbl 1060.30053
Full Text: DOI Euclid


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