Journal of the Mathematical Society of Japan

A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds

Atsushi ATSUJI

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We show that a second main theorem of Nevanlinna theory holds for meromorphic functions on general complete Kähler manifolds. It is well-known in classical Nevanlinna theory that a meromorphic function whose image grows rapidly enough can omit at most two points. Our second main theorem implies this fact holds for meromorphic functions on general complete Kähler manifolds.

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J. Math. Soc. Japan Volume 60, Number 2 (2008), 471-493.

First available in Project Euclid: 30 May 2008

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Zentralblatt MATH identifier

Primary: 32H30: Value distribution theory in higher dimensions {For function- theoretic properties, see 32A22}
Secondary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Nevanlinna theory Brownian motion on Kähler manifolds Kähler diffusion value distribution theory for meromorphic functions


ATSUJI, Atsushi. A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds. J. Math. Soc. Japan 60 (2008), no. 2, 471--493. doi:10.2969/jmsj/06020471.

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  • A. Atsuji, Nevanlinna theory via stochastic calculus, J. Funct. Anal., 132 (1995), 473–510.
  • A. Atsuji, A second main theorem of Nevanlinna theory for meromorphic functions on complex submanifolds in $\ce^n$, submitted.
  • R. F. Bass, Probabilistic Techniques in Analysis., Springer, New York, 1995.
  • J. R. Baxter and G. A. Brosamler, Energy and the law of iterated logarithm, Math. Scand., 38 (1976), 115–136.
  • I. Chavel, Isoperimetric inequalities, Cambridge tracts in mathematics 145, Cambridge university press, Cambridge 2001.
  • B. Davis, Picard's theorem and Brownian motion, Trans. Amer. Math. Soc., 213 (1975), 353–361.
  • B. Davis, Brownian motion and analytic functions, Ann. Prob., 7 (1979), 913–932.
  • R. E. Green and H. Wu, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, 25 (1975), 215–235.
  • P. A. Griffiths, Entire holomorphic mappings in one and several complex variables, Ann. Math. Stud., 85, Princeton University Press, Princeton, 1976.
  • W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs Clarendon Press, Oxford, 1964.
  • W. K. Hayman, Subharmonic functions, 2, Academic press, London-San Diego, 1989.
  • N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Second edition, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
  • K. Ito and H. P. McKean, Diffusion processes and their sample paths, Springer-Verlag, Berlin-New York, 1974.
  • A. Kasue, Applications of Laplacian and Hessian comparison theorems, Geometry of Geodesics and Related Topics, Adv. Stud. Pure Math., 3 (1984), 333–386.
  • H. Kumura, On the intrinsic ultracontractivity for compact manifolds with boundary, Kyushu J. Math., 57 (2003), 29–50.
  • P. Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math., 99 (1990), 579–600.
  • Y. C. Lu, Holomorphic mappings of complex manifolds, J. Diff. Geom., 2 (1968), 299–312.
  • J. Noguchi and T. Ochiai, Geometric function theory in several complex variables, Translations of Mathematical Monographs, 80, Amer. Math. Soc., Providence, RI, 1990.
  • W. Stoll, Value distribution on parabolic spaces, Lecture Notes in Math., 600, Springer-Verlag, Berlin-New York, 1977.
  • M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959.
  • J. Vauthier, Processus projeté, Comparaison avec une diffusion, C. R. Acad. Sci. Paris Sér. A–B, 285 (1977), A569–A571.
  • H. Wu, Mappings of Riemann surfaces (Nevanlinna theory), Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1966, 480–532.