Open Access
Sept. 2005 Dependance of Dirichlet integrals upon lumps of Riemann surfaces
Mitsuru Nakai
Proc. Japan Acad. Ser. A Math. Sci. 81(7): 131-133 (Sept. 2005). DOI: 10.3792/pjaa.81.131


Take a simple arc $\gamma$ in an open Riemann suface $R$ carrying a nonconstant harmonic function $u$ with finite Dirichlet integral $D(u;R)$. Form a Riemann surface $R_{\gamma}$ with lump $\widehat{\mathbf{C}}\setminus\gamma$ by pasting $R\setminus\gamma$ with $\widehat{\mathbf{C}}\setminus\gamma$ crosswise along $\gamma$, i.e. $R_{\gamma}:=(R\setminus\gamma) \utimes{\gamma}(\widehat{\mathbf{C}}\setminus\gamma)$, and the transplant $u_{\gamma}$ of $u$ on $R$ to $R_{\gamma}$ characterized 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. We are interested in the comparison of $D(u_{\gamma};R_{\gamma})$ with $D(u;R)$ when we take a variety of choices of pasting arcs $\gamma$ in $R$, and we will prove that $D(u_{\gamma};R_{\gamma})<D(u;R)$ for any $u$ level arc $\gamma$ in $R$, $D(u_{\gamma};R_{\gamma})>D(u;R)$ for any $u$ conjugate level arc $\gamma$ in $R$, and as a consequence of these two facts there is a nondegenerate arc $\gamma$ (i.e. not a point arc $\gamma$) in $R$ such that $D(u_{\gamma};R_{\gamma})=D(u;R)$.


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Mitsuru Nakai. "Dependance of Dirichlet integrals upon lumps of Riemann surfaces." Proc. Japan Acad. Ser. A Math. Sci. 81 (7) 131 - 133, Sept. 2005.


Published: Sept. 2005
First available in Project Euclid: 3 October 2005

zbMATH: 1108.31002
MathSciNet: MR2172603
Digital Object Identifier: 10.3792/pjaa.81.131

Primary: 31A15 , 31C15
Secondary: 30C85 , 30F15

Keywords: Conjugate level arc , Dirichlet integral , level arc , pasting arc , Riemann surface with lump , Royden decomposition

Rights: Copyright © 2005 The Japan Academy

Vol.81 • No. 7 • Sept. 2005
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