Embedding open Riemann surfaces in Riemannian manifolds

  • Seok-Ku Ko


For the Riemann surface of the topological type, we can get a conformai model in orientable Riemannian manifolds. We will prove that there is a conformally equivalent model in orientable Riemannian manifolds for a given open Riemann surface. To end up we utilize Garsia 's Continuity lemma and Brouwer's Fixed Point lemma along with the Teichmüller theory.

Math Subject Classifications


Key Words and Phrases

embedding open Riemann surface Teichmüller space conformai deformation 


  1. [1]
    Abikoff, W. The real analytic theory of Teichmüller space,Lecture Notes,820, Springer-Verlag, Berlin, 1980.Google Scholar
  2. [2]
    Abikoff, W.Unpublished notes, Springer, 1986.Google Scholar
  3. [3]
    Ahlfors, L. On quasiconformal mappings,J. d'Anal. Math.,3(1), 1–58, (1953–1954).MathSciNetCrossRefGoogle Scholar
  4. [4]
    Ahlfors, L. and Bers, L. Riemann's mapping theorem for variable metrics,Ann. Math.,72(2), (1960).Google Scholar
  5. [5]
    Gardiner, F.Teichmüller Theory and Quadratic Differentials, John Wiley & Sons, New York, 1987.MATHGoogle Scholar
  6. [6]
    Garsia, A.M. An embedding of closed Riemann surfaces in euclidean Space,Comm. Math. Helv.,35, 93–110, (1961).CrossRefMathSciNetGoogle Scholar
  7. [7]
    Klein, F.Gesammelte Math. Abhandlungen, Springer-Verlag, Berlin, 1973.Google Scholar
  8. [8]
    Ko, S.Embedding Riemann Surfaces in Riemannian Manifolds, University of Connecticut Dissertation, 1989.Google Scholar
  9. [9]
    Ko, S. Teichmüller Theory, Its New and Old,Hagsulzi,35, Kon-Kuk University, 1991.Google Scholar
  10. [10]
    Ko, S. Teichmüller theory and its applications,Proc. Pure Math., Workshop at Mokpo University, July, 1991.Google Scholar
  11. [11]
    Ko, S. Embedding bordered Riemann surfaces in riemannian manifolds,J. Korean Math. Soc.,30(2), (1993).Google Scholar
  12. [12]
    Lehner, J.Discontinuous Groups and Automorphic Functions, AMS, Providence, RI, 1964.MATHGoogle Scholar
  13. [13]
    Lehto, O. and Virtanen, K.Quasiconformal Mappings, Springer-Verlag, Berlin, 1965.Google Scholar
  14. [14]
    Nag, S.The Complex Analytic Theory of Teichmüller Spaces, Wiley-Interscience, New York, 1988.MATHGoogle Scholar
  15. [15]
    Nash, J. C 1-isometric embeddings,Ann. Math.,60, 383–396, (1954).CrossRefMathSciNetGoogle Scholar
  16. [16]
    Rüedy, R. Embeddings of open Riemann surfaces,Comm. Math. Helv.,46, (1971).Google Scholar
  17. [17]
    Rüedy, R. Deformations of embedded Riemann surfaces,Ann. Math. Studies,66, (1971).Google Scholar
  18. [18]
    Rüedy, R. Embettung Riemannscher Fläschen in den dreidimensionalen euklidischen Raum,Comm. Math. Helv.,43, 417–442, (1968).CrossRefMATHGoogle Scholar
  19. [19]
    Teichmüller, O. Beweis der analytischen Abhängigkeit des konformen Moduls einer analytischen Ringfläschar von den Parametern,Deutsche Math.,7, 309–336, (1944).MathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1999

Authors and Affiliations

  1. 1.Department of Applied Mathematics, College of Natural ScienceKonKuk UniversityChoongbukS. Korea

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