Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 11, 2017

Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds

Atsushi Atsuji EMAIL logo
From the journal Forum Mathematicum

Abstract

We give a second main theorem of Nevanlinna theory on complete non-positively curved Kähler manifolds. Its remainder term depends only on Ricci curvature of the manifolds except for the terms depending only on the characteristic functions.

MSC 2010: 32H30; 58J65

Communicated by Junjiro Noguchi


Award Identifier / Grant number: 24540192

Funding statement: Partially supported by the Grant-in-Aid for Scientific Research (C) 24540192, Japan Society for the Promotion of Science.

References

[1] A. Atsuji, A Casorati–Weierstrass theorem for holomorphic maps and invariant σ-fields of homomorphic diffusions, Bull. Sci. Math. 123 (1999), no. 5, 371–383. 10.1016/S0007-4497(99)00108-6Search in Google Scholar

[2] A. Atsuji, A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds, J. Math. Soc. Japan 60 (2008), 471–493. 10.2969/jmsj/06020471Search in Google Scholar

[3] A. Atsuji, On the number of omitted values by a meromorphic function of finite energy and heat diffusions, J. Geom. Anal. 20 (2010), no. 4, 1008–1025. 10.1007/s12220-010-9131-6Search in Google Scholar

[4] A. Atsuji, The submartingale property and Liouville type theorems, Manuscripta Math. (2016), 10.1007/s00229-016-0907-2. 10.1007/s00229-016-0907-2Search in Google Scholar

[5] R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964. Search in Google Scholar

[6] T. K. Carne, Brownian motion and Nevanlinna theory, Proc. London Math. Soc. (3) 52 (1986), 349–368. 10.1112/plms/s3-52.2.349Search in Google Scholar

[7] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1990. 10.1017/CBO9780511566158Search in Google Scholar

[8] A. Debiard, B. Gaveau and E. Mazet, Theoremes de comparaison en geometrie Riemannienne, Publ. Res. Inst. Math. Sci. Kyoto 12 (1976), 390–425. 10.2977/prims/1195190722Search in Google Scholar

[9] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. 10.1515/9783110889741Search in Google Scholar

[10] R. E. Greene and H. Wu, Function Theory on Manifolds Which Posses a Pole, Lecture Notes in Math. 699, Springer, Berlin, 1979. 10.1007/BFb0063413Search in Google Scholar

[11] A. Grigoryan, Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. 47, American Mathematical Society, Providence, 2009. Search in Google Scholar

[12] W. K. Hayman, Meromorphic Functions, Oxford University Press, Oxford, 1964. Search in Google Scholar

[13] E. P. Hsu, Stochastic Analysis on Manifolds, Grad. Stud. Math. 38, American Mathematical Society, Providence, 2002. 10.1090/gsm/038Search in Google Scholar

[14] A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72. 10.1007/BF02564570Search in Google Scholar

[15] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland Math. Library 24, North-Holland Publishing, Amsterdam, 1989. Search in Google Scholar

[16] K.-T. Kim and H. Lee, Schwarz’s Lemma from a Differential Geometric Viewpoint, World Scientific, Hackensack, 2010. 10.1142/7944Search in Google Scholar

[17] P. Li and S. T. Yau, Curvature and holomorphic mappings of complete Kähler manifolds, Compos. Math. 73 (1990), 125–144. Search in Google Scholar

[18] Y-C. Lu, Holomorphic mappings of complex manifolds, J. Differential Geom. 2 (1968), 299–312. 10.4310/jdg/1214428442Search in Google Scholar

[19] J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, American Mathematical Society, Providence, 1997. Search in Google Scholar

[20] S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique, Progr. Math. 266, Birkhäuser, Basel, 2008. Search in Google Scholar

[21] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, 1991. 10.1007/978-3-662-21726-9Search in Google Scholar

[22] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Somerville, 1994. Search in Google Scholar

[23] L. Schwartz, Semi-martingales sur des variétés analytiques complexes, Lecture Notes in Math. 780, Springer, Berlin, 1980. 10.1007/BFb0096133Search in Google Scholar

[24] W. Stoll, Value Distribution on Parabolic Spaces, Lecture Notes in Math. 600, Springer, Berlin, 1977. 10.1007/BFb0062904Search in Google Scholar

[25] W. Stoll, Value Distribution Theory for Meromorphic Maps, Vieweg, Braunschweig, 1985. 10.1007/978-3-663-05292-0Search in Google Scholar

Received: 2016-02-06
Revised: 2017-02-16
Published Online: 2017-05-11
Published in Print: 2018-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.12.2022 from https://www.degruyter.com/document/doi/10.1515/forum-2016-0032/html
Scroll Up Arrow