Proceedings of the Japan Academy, Series A, Mathematical Sciences

Evans-Selberg potential on planar domains

Robert Xin Dong

Full-text: Open access


We provide explicit formulas of Evans kernels, Evans-Selberg potentials and fundamental metrics on potential-theoretically parabolic planar domains.

Article information

Proc. Japan Acad. Ser. A Math. Sci. Volume 93, Number 4 (2017), 23-26.

First available in Project Euclid: 31 March 2017

Permanent link to this document

Digital Object Identifier

Primary: 30F15: Harmonic functions on Riemann surfaces
Secondary: 31A05: Harmonic, subharmonic, superharmonic functions 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 30F20: Classification theory of Riemann surfaces

Evans-Selberg potential Evans kernel potential-theoretically parabolic Riemann surface Green function Green kernel fundamental metric


Dong, Robert Xin. Evans-Selberg potential on planar domains. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 4, 23--26. doi:10.3792/pjaa.93.23.

Export citation


  • S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Graduate Texts in Mathematics, 137, Springer, New York, 1992.
  • R. P. Boas, Invitation to complex analysis, 2nd ed., MAA Textbooks, Math. Assoc. America, Washington, DC, 2010.
  • R. Courant and D. Hilbert, Methoden der mathematischen Physik. I, dritte Auflage, Springer, Berlin, 1968.
  • R. X. Dong, Suita conjecture for a punctured torus, in New Trends in Analysis and Interdisciplinary Applications: Selected Contributions of the 10th ISAAC Congress (Macau, 2015), 199–206, Trends in Mathematics, Birkhäuser/Springer, Basel, 2017.
  • G. C. Evans, Potentials and positively infinite singularities of harmonic functions, Monatsh. Math. Phys. 43 (1936), no. 1, 419–424.
  • Z. Kuramochi, Mass distributions on the ideal boundaries of abstract Riemann surfaces. I, Osaka Math. J. 8 (1956), 119–137.
  • N. Mok, The Serre problem on Riemann surfaces, Math. Ann. 258 (1981/82), no. 2, 145–168.
  • M. Nakai, On Evans potential, Proc. Japan Acad. 38 (1962), 624–629.
  • M. Nakai, On Evans' kernel, Pacific J. Math. 22 (1967), 125–137.
  • T. Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, 28, Cambridge Univ. Press, Cambridge, 1995.
  • J. W. Robbin, The Uniformization Theorem, robbin/951dir/uniformization.pdf.
  • H. L. Selberg, Über die ebenen Punktmengen von der Kapazität Null, Avh. Norske Videnskaps-Akad, Oslo I Math.-Natur. 10 (1937).
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer, New York, 1970.
  • L. Sario and K. Noshiro, Value distribution theory, In collaboration with Tadashi Kuroda, Kikuji Matsumoto and Mitsuru Nakai, D. Van Nostrand Co., Inc., Princeton, NJ, 1966.
  • A. Schuster and D. Varolin, Interpolation and sampling for generalized Bergman spaces on finite Riemann surfaces, Rev. Mat. Iberoam. 24 (2008), no. 2, 499–530.