Holomorphic mappings of once-holed tori, II

  • Makoto MasumotoEmail author


In our previous work [13], for a given Riemann surface Y0 with marked handle, we investigated geometric properties of the set of marked once-holed tori X allowing holomorphic mappings of X into Y0. It turned out to be a closed domain with Lipschitz boundary. In the present paper we show that the boundary is not smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.


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The author really appreciates the referee’s careful reading and invaluable comments.


  1. [1]
    H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Fla¨chen, Math. Ann. 120 (1949), 430–461.MathSciNetCrossRefGoogle Scholar
  2. [2]
    M. F. Bourque, The converse of the Schwarz lemma is false, Ann. Acad. Sci. Fenn. 41 (2016), 235–241.MathSciNetCrossRefGoogle Scholar
  3. [3]
    P. Buser, Geometry andSpectra of Compact Riemann Surfaces, Birkhäuser, Boston–Basel–Berlin, 1992.zbMATHGoogle Scholar
  4. [4]
    J. A. Jenkins and N. Suita, On analytic self-mappings of Riemann surfaces II, Math. Ann. 209 (1974), 109–115.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Kahn, K. M. Pilgrim and D. P. Thurston, Conformal surface embeddings and extremal length, preprint, arXiv:1507.05294[math.CV].Google Scholar
  6. [6]
    A. Marden, I. Richards and B. Rodin, Analytic self-mappings of Riemann surfaces, J. Anal. Math. 18 (1967), 197–225.MathSciNetCrossRefGoogle Scholar
  7. [7]
    B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. 10 (1985), 381–386.MathSciNetCrossRefGoogle Scholar
  8. [8]
    W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
  9. [9]
    M. Masumoto, Conformal mappings of a once-holed torus, J. Anal. Math. 66 (1995), 117–136.MathSciNetCrossRefGoogle Scholar
  10. [10]
    M. Masumoto, Once-holed tori embedded in Riemann surfaces, Math. Z. 257 (2007), 453–464.MathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Masumoto, Holomorphic mappings and basic extremal lengths of once-holed tori, Nonlinear Anal. 71 (2009), e1178–e1181.MathSciNetCrossRefGoogle Scholar
  12. [12]
    M. Masumoto, On critical extremal length for the existence of holomorphic mappings of once-holed tori, J. Inequal. Appl. 2013 (2013), article no. 282.Google Scholar
  13. [13]
    M. Masumoto, Holomorphic mappings of once-holed tori, J. Anal. Math. 129 (2016), 69–90.MathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Shiba, The moduli of compact continuations of an open Riemann surface of genus one, Trans. Amer. Math. Soc. 301 (1987), 299–311.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Shiba, The euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems, Kodai Math. J. 16 (1993), 118–137.MathSciNetCrossRefGoogle Scholar
  16. [16]
    K. Strebel, Quadratic Differentials, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1984.CrossRefGoogle Scholar
  17. [17]
    S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2) 109 (1979), 323–351.MathSciNetCrossRefGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsYamaguchi UniversityYamaguchiJapan

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