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Algebraic Criterion on Quasiconformal Equivalence of Riemann Surfaces

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
Mathematical Institute, Nagoya University
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1. Various strides have been done to characterize the conformal structure of Riemann surfaces by the algebraic structure of some appropriate function algebras on them (cf. Bers [2], Rudin [29], Royden [26], [28], Heins [7], Kakutani [12], Wermer [33] etc.). In this paper we discuss, corresponding to the above, the problem to determine the quasiconformal structure of Riemann surfaces by the algebraic structure of some function algebras on them.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 16 , February 1960 , pp. 157 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

[1] Ahlfors, L.: On quasiconformal mappings, J. d’Analyse Math., 3, 1–58, 207–208 (1953-1954).CrossRefGoogle Scholar
[2] Bers, L.: On rings of analytic functions, Bull. Amer. Math. Soc, 54, 311–315 (1948).CrossRefGoogle Scholar
[3] Bers, L.: On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc., 84, 78–84 (1957).CrossRefGoogle Scholar
[4] Bourbaki, N.: Topologie générale, Chap. X (1949).Google Scholar
[5] Čech, E.: On bicompact spaces, Ann., of Math., 38, 823–844 (1937).CrossRefGoogle Scholar
[6] Gelfand, I.: Normierte Ringe, Rec. Math., 9, 3–24 (1941).Google Scholar
[7] Heins, M.: Algebraic structure and conformal mapping, Trans. Amer. Math. Soc., 89, 267–276 (1958).CrossRefGoogle Scholar
[8] Hewitt, E.: Rings of real-valued continuous functions I, Trans. Amer. Math. Soc, 64, 45–99 (1948).CrossRefGoogle Scholar
[9] Ishii, T.: On homomorphisms of the ring of continuous functions onto the real numbers, Proc Japan Acad., 33, 419–423 (1957).Google Scholar
[10] Ishiwata, T.: On the ring of all bounded continuous functions, Sci. Rep. Tokyo Kyoiku Daigaku, 5, 279–280 (1957).Google Scholar
[11] Ishiwata, T.: On locally Q-complete spaces, II, Proc Japan Acad., 35, 263–267 (1959).Google Scholar
[12] Kakutani, S.: On rings of analytic functions, Proc. Michigan Conference on functions of a complex variable (1953).Google Scholar
[13] Loomis, L.: An introduction to abstract harmonic analysis (1953).Google Scholar
[14] Mori, A.: On quasi-conformality and pseudo-analyticity, Trans. Amer. Math. Soc., 84, 56–77 (1957).CrossRefGoogle Scholar
[15] Mori, S.: A remark on a subdomain of a Riemann surface of the class OHD , Proc. Japan Acad., 34, 251–254 (1958).Google Scholar
[16] Mori, S.: On the ideal boundary of a simply connected open Riemann surface (in Japanese), Mem. Research Inst. Sci. and Eng., Ritsumeikan Univ., 2, 1–6 (1957).Google Scholar
[17] Mori, S. and Ota, M.: A remark on the ideal boundary of a Riemann surface, Proc, Japan Acad., 32, 409–411 (1956).Google Scholar
[18] Nakai, M.: On a ring isomorphism induced by quasiconformal mappings, Nagoya Math. J., 14, 201–221 (1959).CrossRefGoogle Scholar
[19] Nakai, M.: A function algebra on Riemann surfaces, Nagoya Math. J., 15, 1–7 (1959).CrossRefGoogle Scholar
[20] Nakai, M.: Purely algebraic characterization of Quasiconformality, Proc. Japan Acad., 35, 440–443 (1959)Google Scholar
[21] Pfluger, A.: Quasiconforme Abbildungen und logarithmische Kapazität, Ann. l’Inst. Fourier, 2, 69–80 (1951).Google Scholar
[22] Pfluger, A.: Über die Äquivalenz der geometrischen und der analytischen Definition quasikonformer Abbildungen, Comment. Math. Helvet, 33, 23–33 (1959).CrossRefGoogle Scholar
[23] Pursell, E.: An algebraic characterization of fixed ideals in certain function rings, Pacific J. Math., 5, 963–969 (1955).CrossRefGoogle Scholar
[24] Royden, H.: The ideal boundary of an open Riemann surface, Annals of Mathematics Studies, 30, 107–109 (1953).Google Scholar
[25] Royden, H.: A property of quasi-conformal mapping, Proc. Amer. Math. Soc., 5, 266–269 (1954).CrossRefGoogle Scholar
[26] Royden, H.: Rings of analytic and meromorphic functions, Trans. Amer. Math. Soc., 83, 269–276 (1956).CrossRefGoogle Scholar
[27] Royden, H.: Open Riemann surfaces, Ann. Acad. Sci. Fenn., A.I. 249/5 (1958).Google Scholar
[28] Royden, H.: Rings of meromorphic functions, Seminars on analytic functions, Inst. for Advanced study, 2, 273–285 (1958).Google Scholar
[29] Rudin, W.: Some theorems on bounded analytic functions, Trans. Amer. Math. Soc., 78, 338–342 (1955).CrossRefGoogle Scholar
[30] Saks, S.: Theory of the integral, 2nd ed., Warsaw (1937).Google Scholar
[31] Shanks, M.: Rings of functions on locally compact spaces, Bull. Amer. Math. Soc., 57, 295 (1951).Google Scholar
[32] Shirota, T.: A generalization of a theorem of I. Kaplansky, Osaka Math. J., 4, 121–132 (1952).Google Scholar
[33] Wermer, J.: The maximum principle for bounded functions, Ann. of Math., 69, 598–604 (1959).CrossRefGoogle Scholar
[34] Yujobo, Z.: On absolutely continuous functions of two or more variables in the Tonelli sense and quasiconformal mappings in the A. Mori sense, comment. Math. Univ. st. Pauli, 4, 67–92 (1955).Google Scholar