Uniqueness Results for Holomorphic Mappings on the Disc

  • Min RuEmail author
  • Richard Walden


This paper derives several unicity results for a class of holomorphic mappings from the disc into compact Riemann surfaces as well as into the complex projective space n(). This is done by using the Nevanlinna theory for holomorphic maps where the source is a disc developed by Ru-Sibony (to appear).


Holomorphic mapping Riemann surfaces Nevanlinna theory 

Mathematics Subject Classification (2010)

32H30 30D35 



  1. 1.
    Cartan, H.: Sur les zeros des combinaisions linearires de p fonctions holomorpes donnees. Mathematica(Cluj) 7, 80–103 (1933)Google Scholar
  2. 2.
    Chern, S.S.: Complex analytic mappings of Riemann surfaces. I. Am. J. Math. 82, 323–337 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z.H., Yan, Q.: Uniqueness theorem of meromorphic mappings into N() sharing 2N + 3 hyperplanes regardless of multiplicities. Internat. J. Math. 20(6), 717–726 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fujimoto, H.: The uniqueness problem for meromorphic maps into complex projective space. Nagoya Math. J. 58, 1–23 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Giang, H.H., Quynh, L.N., Quang, S.D.: Uniqueness theorems for meromorphic mappings sharing few hyperplanes. J. Math. Anal. Appl. 393(2), 445–456 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Miranda, R.: Algebraic Curves and Riemann Surfaces Graduate Studies in Mathematics, vol. 5. Am. Math. Soc., Providence (1995)Google Scholar
  7. 7.
    Nevanlinna, R.: Zur theorie der meromorphen funktionen. Acta Math. 46(1–2), 1–99 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Noguchi, J., Winkelmann, J.: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Springer, Tokyo (2014)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ru, M.: Nevanlinna Theory and its Relation to Diophantine Approximation. World Scientific (2001)Google Scholar
  10. 10.
    Ru, M., Sibony, N.: The second main theorem in the hyperbolic case. Math. Ann. (to appear)Google Scholar
  11. 11.
    Ru, M., Ugur, G.: Uniqueness results for algebraic and holomorphic curves into \(\phantom {\dot {i}\!}\mathbb {P}^{n}(\mathbb {C})\). Internat J. Math. 28(9), 27 pp. (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sauer, A.: Uniqueness theorems for holomorphic functions on compact Riemann surfaces. New Zealand J. Math. 30(2), 188–181 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Schmid, E.M.: Some theorems on value distributions of meromorphic functions. Math. Z. 120, 61–92 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shabat, B.V.: Distribution of Values of Holomorphic Mappings. Translated from the Russian by J. R. King. Translation edited by Lev J. Leifman Translations of Mathematical Monographs, vol. 61. American Mathematical Society, Providence (1985)Google Scholar
  15. 15.
    Vojta, P.: Diophantine Approximations and Value Distribution Theory. Lecture Notes in Math, vol. 1239. Springer, Berlin (1987)Google Scholar
  16. 16.
    Xu, Y., Ru, M.: Uniqueness theorem for algebraic curves on compact Riemann surfaces. Sci. China Ser. 50(5), 683–688 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

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