Dependence of Dirichlet integrals upon lumps of Riemann surfaces.
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 \(D(u_\gamma;R_\gamma)=D(u;R)\).
scientific article; zbMATH DE number 5017481