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Sario’s Potentials and Analytic Mappings*

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
University of California, Los Angeles and Nagoya University
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In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 29 , March 1967 , pp. 93 - 101
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

Footnotes

*

The work was sponsored by the U.S. Army Research Office Durham, Grant DA-AROD-31-124-G 742, University of California, Los Angeles.

References

[1] Ahlfors, L. V. and Sario, L., Riemann surfaces, Princeton Univ. Press, Princeton, 1960, 382 pp.Google Scholar
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[3] Nakai, M., On Evans potential, Proc. Japan Acad., 38 (1962), 624–629.Google Scholar
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[6] Noshiro, K., Cluster sets, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960, 135 pp.Google Scholar
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[8] Sario, L., Value distribution under analytic mappings of arbitrary Riemann surfaces, Acta Math., 109 (1963), 1–10.Google Scholar
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[10] Sario, L., A theorem on mappings into Riemann surfaces of infinite genus, Trans. Amer. Math. Soc., 117 (1965), 276–284.Google Scholar