Cluster Sets pp 90-109 | Cite as

Conformal mapping of Riemann surfaces

  • Kiyoshi Noshiro
Part of the Ergebnisse der Mathematik und Ihrer Grenzgebiete book series (MATHE2, volume 28)


Throughout this chapter, we shall consider Riemann surfaces in the sense of Weyl and Radó.


Harmonic Function Riemann Surface Conformal Mapping Harmonic Measure Parabolic Type 
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  1. 1.
    For example, some Schwarz ’s triangle functions. See also Noshiro [3], p. 154.Google Scholar
  2. 1.
    Sario [1] called e μ the modulus of G. This distinction is not essential.Google Scholar
  3. 1.
    Sario [3] remarked that a graph K of finite length can be constructed by a suitable choice of an exhaustion of F, in the case when F is simply connected and of parabolic type.Google Scholar
  4. 2.
    Noshiro [9], p. 76.Google Scholar
  5. 2.
    Yujôbô [2], Tsuji [13].Google Scholar
  6. 1.
    Cf. Noshiro [4] for some related results.Google Scholar
  7. 2.
    The importance of such a class was already pointed out by Sario [2].Google Scholar
  8. 1.
    For the sufficiency, the writer overs ti Kuroda Google Scholar
  9. 1.
    Cf. Constantinescu-Cornea [1], p. 194.Google Scholar
  10. 1.
    In a special case, this lemma was proved by Constantinescu-Cornea ([1], p. 189).Google Scholar
  11. 1.
    The following proof is due to Matsumoto [2].Google Scholar
  12. 1.
    To prove (i), Mori uses Rademacher-Stepanoff’s theorem (Saks [1], p. 310).Google Scholar
  13. 2.
    The proofs of Strebel [1] and Pfluger [5] for (iii) are simpler than that of Mori [9]. From (iii) and (iv), it follows that w = T (z) is absolutely continuous in the sense of Tonelli in D (Saks [1], p. 169).Google Scholar
  14. 1.
    Ahlfors gave a simple proof for (v) in his lecture at Osaka University in 1955.Google Scholar
  15. 2.
    This theorem means that quasiconformality is a local property.Google Scholar
  16. 3.
    The proof is due to Mori [9].Google Scholar
  17. 4.
    Cf. Sakai [1], Yûjôbô [4, 5].Google Scholar
  18. 1.
    The proof of Theorem 5, § 4, II is essentially based upon the theory of functions of class (U) in Seidel’s sense. It is very important to remark that this method is not available in the case of KPA functions as we shall see in the next paragraph.Google Scholar
  19. 2.
    Cf. Noshiro [9].Google Scholar
  20. 1.
    Cf. Kaplan [1,2].Google Scholar
  21. 2.
    Compared with Theorem 5, we see that the set E z is of positive capacity.Google Scholar
  22. 3.
    Cf. Shibata [2].Google Scholar
  23. 3.
    The following proof is due to Matsumoto.Google Scholar

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© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

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  • Kiyoshi Noshiro

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