Journal of the Mathematical Society of Japan

Conformal invariants defined by harmonic functions on Riemann surfaces

Hiroshige SHIGA

Full-text: Open access


In this paper, we consider conformal invariants defined by various spaces of harmonic functions on Riemann surfaces. The Harnack distance is a typical one. We give sharp inequalities comparing those invariants with the hyperbolic metric on the Riemann surface and we determine when equalities hold. We also describe the Harnack distance in terms of the Martin compactification and discuss some properties of the distance.

Article information

J. Math. Soc. Japan, Volume 68, Number 1 (2016), 441-458.

First available in Project Euclid: 25 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 30C40: Kernel functions and applications 30F60: Teichmüller theory [See also 32G15] 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems

Harnack distance harmonic Hardy space hyperbolic distance


SHIGA, Hiroshige. Conformal invariants defined by harmonic functions on Riemann surfaces. J. Math. Soc. Japan 68 (2016), no. 1, 441--458. doi:10.2969/jmsj/06810441.

Export citation


  • S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd ed., Graduate Texts in Mathematics, 137, Springer-Verlag, 2001.
  • C. Constantinscu and A. Cornea, Ideale Ränder Riemannscher Flächen, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963.
  • D. A. Herron, The Harnack and other conformally invariant metrics, Kodai Math. J., 10 (1987), 9–19.
  • D. A. Herron and D. Minda, Comparing invariant distances and conformal metrics on Riemann surfaces, Israel J. Math., 122 (2001), 207–220.
  • F. L. Lárusson, A Wolff–Denjoy theorem for infinitely connected Riemann surfaces, Proc. Amer. Math. Soc., 124 (1996), 2745–2750.
  • K. Oikawa, A constant related to harmonic functions, Japanese J. Math., 29 (1959), 111–113.
  • H. Tanaka, On Harnack's pseudo-distance, Hokkaido Math. J., 6 (1977), 302–305.