Proceedings of the Japan Academy, Series A, Mathematical Sciences

Dependance of Dirichlet integrals upon lumps of Riemann surfaces

Mitsuru Nakai

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Abstract

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

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 7 (2005), 131-133.

Dates
First available in Project Euclid: 3 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1128346017

Digital Object Identifier
doi:10.3792/pjaa.81.131

Mathematical Reviews number (MathSciNet)
MR2172603

Zentralblatt MATH identifier
1108.31002

Subjects
Primary: 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 31C15: Potentials and capacities
Secondary: 30C85: Capacity and harmonic measure in the complex plane [See also 31A15] 30F15: Harmonic functions on Riemann surfaces

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

Citation

Nakai, Mitsuru. Dependance of Dirichlet integrals upon lumps of Riemann surfaces. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 7, 131--133. doi:10.3792/pjaa.81.131. https://projecteuclid.org/euclid.pja/1128346017


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References

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