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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Holomorphic curves with shift-invariant hyperplane preimages
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by Rodney Halburd, Risto Korhonen and Kazuya Tohge PDF
Trans. Amer. Math. Soc. 366 (2014), 4267-4298 Request permission


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
  • MR Author ID: 330280
  • Email:
  • Risto Korhonen
  • Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
  • MR Author ID: 702144
  • Email:
  • Kazuya Tohge
  • Affiliation: School of Electrical and Computer Engineering, College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan
  • MR Author ID: 253733
  • Email:
  • 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.
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 4267-4298
  • MSC (2010): Primary 32H30; Secondary 30D35
  • DOI:
  • MathSciNet review: 3206459