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 \(D(u_\gamma;R_\gamma)=D(u;R)\).


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
Full Text: DOI


[1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory , 2nd ed., Springer, New York, 2001. · Zbl 0959.31001
[2] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen , Springer, Berlin, 1963. · Zbl 0112.30801
[3] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations , Oxford Univ. Press, New York, 1993. · Zbl 0780.31001
[4] M. Nakai, Types of pasting arcs in two sheeted spheres , in Potential theory ( Matsue , 2004), Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, 2005. (To appear).
[5] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Springer, New York, 1970. · Zbl 0199.40603
[6] J.L. Schiff, Normal families , Springer, New York, 1993. · Zbl 0770.30002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.