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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Holomorphic curves with shift-invariant hyperplane preimages


Authors: Rodney Halburd, Risto Korhonen and Kazuya Tohge
Journal: Trans. Amer. Math. Soc. 366 (2014), 4267-4298
MSC (2010): Primary 32H30; Secondary 30D35
Published electronically: March 24, 2014
MathSciNet review: 3206459
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Abstract: If $ f:\mathbb{C}\to \mathbb{P}^n$ is a holomorphic curve of hyper-order less than one for which $ 2n+1$ hyperplanes in general position have forward invariant preimages with respect to the translation $ \tau (z)= z+c$, then $ f$ is periodic with period  $ c\in \mathbb{C}$. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.


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Additional Information

Rodney Halburd
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email: r.halburd@ucl.ac.uk

Risto Korhonen
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: risto.korhonen@uef.fi

Kazuya Tohge
Affiliation: School of Electrical and Computer Engineering, College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan
Email: tohge@se.kanazawa-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05949-7
Keywords: Holomorphic curve, Casorati determinant, difference operator, Borel's theorem, Nevanlinna theory, Cartan's second main theorem
Received by editor(s): March 26, 2009
Received by editor(s) in revised form: February 21, 2011, October 30, 2011, and August 30, 2012
Published electronically: March 24, 2014
Additional Notes: This research was supported in part by the Academy of Finland Grant #112453, #118314 and #210245, a grant from the EPSRC, the Isaac Newton Institute for Mathematical Sciences, the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (C) #19540173, #22540181, and a project grant from the Leverhulme Trust.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.