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Dependence of Dirichlet integrals upon lumps of Riemann surfaces. (English) Zbl 1108.31002

The author takes a simple arc \(\gamma\) in an open Riemann surface \(R\) carrying a non-constant harmonic function \(u\) with a finite Dirichlet integral \(D(u;R)\). Then he forms a Riemann surface \(R_\gamma\) with lump \(\widehat C\setminus\gamma\) by pasting \(R\setminus\gamma\) with \(\widehat C\setminus\gamma\) crosswise along \(\gamma\), i.e., \(R_\gamma:=(R\setminus\gamma)\cup_\gamma (\widehat C\setminus \gamma)\), and he transplants \(u_\gamma\) of \(u\) on \(R\) to \(R_\gamma\) characterizing by its being harmonic on \(R_\gamma\) with \(D(u_\gamma;R_\gamma)<+ \infty\) and \(u_\gamma =u\) at the ideal boundary of \(R_\gamma\) and hence of \(R\) in a suitable sense. Among other things, he proves that there is a non-degenerate arc \(\gamma\) (i.e., not a point arc \(\gamma)\) in \(R\) such that \(D(u_\gamma;R_\gamma)=D(u;R)\).

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31C15 Potentials and capacities on other spaces
30C85 Capacity and harmonic measure in the complex plane
30F15 Harmonic functions on Riemann surfaces
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References:

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