Proceedings of the Japan Academy, Series A, Mathematical Sciences

Dependance of Dirichlet integrals upon lumps of Riemann surfaces

Mitsuru Nakai

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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

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

First available in Project Euclid: 3 October 2005

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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

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


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.

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  • S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, 2nd ed., Springer, New York, 2001.
  • C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Springer, Berlin, 1963.
  • J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Univ. Press, New York, 1993.
  • 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).
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Springer, New York, 1970.
  • J.L. Schiff, Normal families, Springer, New York, 1993.