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General Value Distribution Theory*

Published online by Cambridge University Press:  22 January 2016

Leo Sario
Leo Sario*
Affiliation:
University of California, Los Angeles
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We shall introduce the main theorems of value distribution theory in the most general case of complex dimension one: analytic mappings of arbitrary Riemann surfaces into arbitrary Riemann surfaces. The case of mappings of arbitrary Riemann surfaces into closed Riemann surfaces was discussed in [41]. Earlier literature on analytic mappings is listed in the Bibliography.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 23 , December 1963, pp. 213 - 229
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

Footnotes

*

Two invited hour lectures delivered at Stanford University on May 27 and 30, 1963. The work was sponsored by the U. S. Army Research Office (Durham), Grant D A-ARO (D)-31-124-G40, University of California, Los Angeles.

References

[1] Ahlfors, L., Über eine Methode in der Theorie der meromorphen Funktionen, Soc. Sci. Fenn. Comment. Phys.-Math. VIII. 10 (1935), 14 pp.Google Scholar
[2] Ahlfors, L., Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157–194.CrossRefGoogle Scholar
[3] Ahlfors, L., Über die Anwendung differentialgeometrischer Methoden zur Untersuchung von Überlagerungsflächen, Acta Soc. Sci. Fenn. Nova Ser. A. II 6 (1937), 17 pp.Google Scholar
[4] Ahlfors, L. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, (1960), 382 pp.CrossRefGoogle Scholar
[5] Bergman, S., Über meromorphe Funktionen von zwei komplexen Veränder lichen, Compositio Math. 6 (1939), 305–335.Google Scholar
[6] Bergman, S., On meromorphic functions of several complex variables, Abstract of short communications, p. 63, Internat. Congr. Math., Stockholm, 1962.Google Scholar
[7] Bergman, S., On value distribution of meromorphic functions of two complex variables, Studies in Mathematical Analysis and Related Topics, Stanford University Press, Stanford, Calif., 1962, pp.25–37.Google Scholar
[8] Chern, S., Complex analytic mappings of Riemann surfaces I, Amer. J. Math. 82 (1960), 323–337.CrossRefGoogle Scholar
[9] Chern, S., The integrated form of the first main theorem for complex analytic mappings in several complex variables, Ann. of Math. 71 (1960), 536–552.CrossRefGoogle Scholar
[10] Chern, S., Minimal surfaces in an Euclidian space of n dimensions (to appear).Google Scholar
[11] Emig, P., Meromorphic functions and the capacity function on abstract Riemann surfaces, Doctoral dissertation, University of California, Los Angeles, 1962, 81 pp.Google Scholar
[12] Evans, G. C., Potential and positively infinite singularities of harmonic functions, Monatsh. Math. Phys. 43 (1936), 419–424.CrossRefGoogle Scholar
[13] Goldstein, M., L- and K-kernels on an arbitrary Riemann surface, Doctoral dissertation, University of California, Los Angeles, (1963), 71 pp.Google Scholar
[14] Hällström, G. af, Über meromorphe Funktionen mit mehrfach zusammenhängenden Existenzgebieten, Acta Acad. Aboensis, Math. Phys. XII. 8 (1939), 5–100.Google Scholar
[15] Heins, M., Riemann surfaces of infinite genus, Ann. of Math. 55 (1952), 296–317.CrossRefGoogle Scholar
[16] Heins, M., Lindelöfian maps, Ann. of Math. (2) 62 (1955), 18–46.CrossRefGoogle Scholar
[17] Heins, M., Functions of bounded characteristic and Lindelöfian maps, Proc. Internat. Congress Math. 1958, 376–388, Cambridge Univ. Press, New York, 1960.Google Scholar
[18] Kunugui, K., Sur l’allure d’une fonction analytique uniforme au voisinage d’un point frontière de son domaine de definition, Japan. J. Math. 18 (1942), 1–39.CrossRefGoogle Scholar
[19] Kuramochi, Z., Evans’ theorem on abstract Riemann surfaces with null boundaries, I-II, Proc. Jap. Acad. 32 (1956), 1–9.Google Scholar
[20] Kuramochi, Z., On covering surfaces, Osaka Math. J. 5 (1953), 155–201.Google Scholar
[21] Levine, H., A theorem on holomorphic mappings into complex projective space, Ann. of Math. 71 (1960), 529–535.CrossRefGoogle Scholar
[22] Matsumoto, K., On exceptional values of meromorphic functions with the set of singularities of capacity zero, Nagoya Math. J. 18 (1961), 171–191.CrossRefGoogle Scholar
[23] Myrberg, L., Uber meromorphe Funktionen und Kovarianten auf Riemannschen Flächen, Ann. Acad. Sci. Fenn. A. I. 244 (1957), 18 pp.Google Scholar
[24] Nevanlinna, F., Über die Anwendung einer Klasse von uniformisierenden Transzendenten zur Untersuchung der Wertverteilung analytischer Funktionen, Acta Math. 50 (1927), 159–188.CrossRefGoogle Scholar
[25] Nevanlinna, R., Zur Theorie der meromorphen Funktionen, Acta Math. 46 (1925), 1–99.CrossRefGoogle Scholar
[26] Nevanlinna, R., Eindeutige analytische Funktionen, Springer, Berlin-Göttingen-Heidelberg, 1953, 379 pp.CrossRefGoogle Scholar
[27] Nakai, M., On Evans potential, Proc. Jap. Acad. 38 (1962), 624–629.Google Scholar
[28] Noshiro, K., Open Riemann surfaces with null boundary, Nagoya Math. J. 3 (1951), 73–79.CrossRefGoogle Scholar
[29] Ohtsuka, M., Reading of the paper “On covering surfaces” by Z. Kuramochi, Ch. I, ditto manuscript, 24 pp.Google Scholar
[30] Osserman, R., Proof of a conjecture by Nirenberg, Comm. Pure Appl. Math. 12 (1959), 229–232.CrossRefGoogle Scholar
[31] Parreau, M., Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Ann. Inst. Fourier 3 (1951), 103–197.CrossRefGoogle Scholar
[32] Rao, K. V. R.. Lindeiöfian maps and positive harmonic functions, Doctoral dissertation, University of California, Los Angeles, April, 1962, 48 pp.Google Scholar
[33] Rao, K. V. R.. Lindeiöfian meromorphic functions, Proc. Amer. Math. Soc. (to appear).Google Scholar
[34] Remoundos, G., Extension aux fonctions algébroides multiformes du théorème de M. Picard et ses generalizations, Mémor. Sci. Math. 23 (1927), 66 pp.Google Scholar
[35] Rodin, B., Reproducing kernels and principal functions, Proc. Amer. Math. Soc. 13 (1962), 982–992.CrossRefGoogle Scholar
[36] Sario, L., Capacity of the bonndary and of a boundary component, Ann. of Math. (2) 59 (1954), 135–144.CrossRefGoogle Scholar
[37] Sario, L., Picard’s great theorem on Riemann surfaces, Amer. Math. Monthly 69 (1962), 598–608.CrossRefGoogle Scholar
[38] Sario, L., Analytic mappings between arbitrary Riemann surfaces (Research announcement), Bull. Amer. Math. Soc, 68 (1962), 633–637.CrossRefGoogle Scholar
[39] Sario, L., Meromorphic functions and conformal metrics on Riemann surfaces, Pacific J. Math. 12 (1962), 1079–1098.CrossRefGoogle Scholar
[40] Sario, L., Islands and peninsulas on arbitrary Riemann surfaces, Trans. Amer. Math. Soc. 106 (1963), 521–532.Google Scholar
[41] Sario, L., Value distribution under analytic mappings of arbitrary Riemann surfaces, Acta Math. 109 (1963), 1–10.CrossRefGoogle Scholar
[42] Sario, L., On locally meromorphic functions with single-valued moduli, Pacific J. Math. 13 (1963), 709–724.CrossRefGoogle Scholar
[43] Sario, L., An integral equation and a general existence theorem for harmonic functions, Comm. Math. Helv. (to appear).Google Scholar
[44] Sario, L., A theorem on mappings into Riemann surfaces of infinite genus, Trans. Amer. Math. Soc. (to appear).Google Scholar
[45] Schwarz, M.-H., Formules apparentées à la formule de Gauss-Bonnet pour certaines applications d’une variété à n dimensions dans une autre, Acta Math. 91 (1954), 189–244.CrossRefGoogle Scholar
[46] Selberg, H., Algebroide Funktionen und Umkehrfunktionen Abelschen Integrale, Avh. Norske Vid.-Acad. Oslo Mat. naturvid. Kl. 8 (1934), 72 pp.Google Scholar
[47] Stoll, W., Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen, Acta Math. 90 (1953), 1–115.CrossRefGoogle Scholar
[48] Tamura, J., Meromorphic functions on open Riemann surfaces, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 9 (1959), 175–186.Google Scholar
[49] Tsuji, M., Existence of a potential function with a prescribed singularity on any Riemann surface, Tôhoku Math. J. (2) 4 (1952), 54–68.CrossRefGoogle Scholar
[50] Tsuji, M., Theory of meromorphic functions on an open Riemann surface with null boundary, Nagoya Math J. 6 (1953), 137–150.CrossRefGoogle Scholar
[51] Tumura, Y., Quelques applications de la théorie de M. Ahlfors, Japan J. Math. 18 (1942), 303–322.CrossRefGoogle Scholar
[52] Ullrich, E., Über den Einfluss der Verzweigtheit einer Algebroide auf ihre Wertverteilung, J. Reine Angew. Math. 167 (1932), 198–220.CrossRefGoogle Scholar
[53] Valiron, G., Fonctions entière d’ordre fini et fonctions méromorphes V, L’Enseignement Math., IIe Série, 5 (1959), 1–28.Google Scholar
[54] Weill, G., Reproducing kernels and orthogonal kernels for analytic differentials on Riemann surfaces, Pacific J. Math. 12 (1962), 729–768.CrossRefGoogle Scholar
[55] Weill, G., Capacity differentials on open Riemann surfaces, Pacific J. Math. 12 (1962), 763–776.CrossRefGoogle Scholar
[56] Weill, G., Some extremal properties of linear combinations of kernels on Riemann surfaces, Pacific J. Math. 12 (1962), 1459–1465.CrossRefGoogle Scholar
[57] Wittich, H., Neuere Untersuchungen über eindeutige analytische Funktionen, Ergebnisse der Mathematik und ihrer Grenzgebiete (n.F. ), Heft 8, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955, 163 pp.Google Scholar