Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2014-05949-7
Published electronically: March 24, 2014
MathSciNet review: 3206459
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), no. 3, 889-905. MR 1759006 (2001g:39003), https://doi.org/10.1088/0951-7715/13/3/321
  • [2] G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • [3] D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, Nevanlinna theory for the $ q$-difference operator and meromorphic solutions of $ q$-difference equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3, 457-474. MR 2332677 (2008f:30072), https://doi.org/10.1017/S0308210506000102
  • [4] W. Bergweiler, A question of Goldberg concerning entire functions with prescribed zeros, J. Anal. Math. 63 (1994), 121-129. MR 1269217 (95b:30038), https://doi.org/10.1007/BF03008421
  • [5] A. Bloch, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Sci. École Norm. Sup. (3) 43 (1926), 309-362 (French). MR 1509274
  • [6] H. Cartan, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, Ann. Sci. École Norm. Sup. (3) 45 (1928), 255-346 (French). MR 1509288
  • [7] H. Cartan, Sur lés zeros des combinaisons linéaires de $ p$ fonctions holomorphes données, Mathematica Cluj 7 (1933), 5-31.
  • [8] W. Cherry and Z. Ye, Nevanlinna's theory of value distribution, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. The second main theorem and its error terms. MR 1831783 (2002h:30030)
  • [9] Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of $ f(z+\eta )$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. MR 2407244 (2009c:30073), https://doi.org/10.1007/s11139-007-9101-1
  • [10] Y.-M. Chiang and S.-J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3767-3791. MR 2491899 (2010c:30044), https://doi.org/10.1090/S0002-9947-09-04663-7
  • [11] Y.-M. Chiang and S. N. M. Ruijsenaars, On the Nevanlinna order of meromorphic solutions to linear analytic difference equations, Stud. Appl. Math. 116 (2006), no. 3, 257-287. MR 2220338 (2006m:39005), https://doi.org/10.1111/j.1467-9590.2006.00343.x
  • [12] J. Clunie, The composition of entire and meromorphic functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 75-92. MR 0271352 (42 #6235)
  • [13] R. J. Evans and I. M. Isaacs, Generalized Vandermonde determinants and roots of unity of prime order, Proc. Amer. Math. Soc. 58 (1976), 51-54. MR 0412205 (54 #332)
  • [14] H. Fujimoto, On holomorphic maps into a taut complex space, Nagoya Math. J. 46 (1972), 49-61. MR 0310274 (46 #9375)
  • [15] H. Fujimoto, On meromorphic maps into the complex projecive space, J. Math. Soc. Japan 26 (1974), 272-288. MR 0346198 (49 #10924)
  • [16] A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; with an appendix by Alexandre Eremenko and James K. Langley. MR 2435270 (2009f:30067)
  • [17] M. L. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89-103. MR 0308433 (46 #7547)
  • [18] M. Green, On the functional equation $ f^{2}=e^{2\phi _{1}}+e^{2\phi _{2}}+e^{2\phi _{3}}\ $ and a new Picard theorem, Trans. Amer. Math. Soc. 195 (1974), 223-230. MR 0348112 (50 #610)
  • [19] M. L. Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43-75. MR 0367302 (51 #3544)
  • [20] G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. (2) 20 (1979), no. 3, 457-466. MR 561137 (80m:30030), https://doi.org/10.1112/jlms/s2-20.3.457
  • [21] G. G. Gundersen, Meromorphic functions that share four values, Trans. Amer. Math. Soc. 277 (1983), no. 2, 545-567. MR 694375 (84g:30028), https://doi.org/10.2307/1999223
  • [22] G. G. Gundersen, Complex functional equations, Complex differential and functional equations (Mekrijärvi, 2000), Univ. Joensuu Dept. Math. Rep. Ser., vol. 5, Univ. Joensuu, Joensuu, 2003, pp. 21-50. MR 1968109 (2004c:39059)
  • [23] G. G. Gundersen and W. K. Hayman, The strength of Cartan's version of Nevanlinna theory, Bull. London Math. Soc. 36 (2004), no. 4, 433-454. MR 2069006 (2005i:30045), https://doi.org/10.1112/S0024609304003418
  • [24] R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477-487. MR 2185244 (2007e:39030), https://doi.org/10.1016/j.jmaa.2005.04.010
  • [25] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478. MR 2248826 (2007e:30038)
  • [26] R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 443-474. MR 2308234 (2008c:39006), https://doi.org/10.1112/plms/pdl012
  • [27] R. G. Halburd and R. J. Korhonen, Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations, J. Phys. A: Math. Theor. 40 (2007), R1-R38.
  • [28] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038 (29 #1337)
  • [29] W. K. Hayman, Warings Problem für analytische Funktionen, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. (1984), 1-13 (1985).
  • [30] A. Hinkkanen, A sharp form of Nevanlinna's second fundamental theorem, Invent. Math. 108 (1992), no. 3, 549-574. MR 1163238 (93c:30045), https://doi.org/10.1007/BF02100617
  • [31] I. Laine and C.-C. Yang, Clunie theorems for difference and $ q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556-566. MR 2377111 (2009b:30063), https://doi.org/10.1112/jlms/jdm073
  • [32] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677 (88f:32065)
  • [33] J. Miles, Quotient representations of meromorphic functions, J. Analyse Math. 25 (1972), 371-388. MR 0350001 (50 #2494)
  • [34] R. Nevanlinna, Zur Theorie der Meromorphen Funktionen, Acta Math. 46 (1925), no. 1-2, 1-99 (German). MR 1555200, https://doi.org/10.1007/BF02543858
  • [35] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes., VII + 174 p. Paris, Gauthier-Villars (Collections de monographies sur la théorie des fonctions) , 1929 (French).
  • [36] M. Ru, Nevanlinna theory and its relation to Diophantine approximation, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1850002 (2002g:11106)
  • [37] S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), no. 2, 1069-1146. MR 1434226 (98m:58065), https://doi.org/10.1063/1.531809
  • [38] S. N. M. Ruijsenaars, On Barnes' multiple zeta and gamma functions, Adv. Math. 156 (2000), no. 1, 107-132. MR 1800255 (2002b:33022), https://doi.org/10.1006/aima.2000.1946
  • [39] P.-M. Wong, H.-F. Law, and P. P. W. Wong, A second main theorem on $ \mathbb{P}^n$ for difference operator, Sci. China Ser. A 52 (2009), no. 12, 2751-2758. MR 2577188 (2010k:32026), https://doi.org/10.1007/s11425-009-0213-5
  • [40] N. Yanagihara, Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), no. 3, 309-326. MR 621536 (82m:39006a)
  • [41] L. Yang, Value distribution theory, Springer-Verlag, Berlin, 1993. Translated and revised from the 1982 Chinese original. MR 1301781 (95h:30039)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32H30, 30D35

Retrieve articles in all journals with MSC (2010): 32H30, 30D35


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: https://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.

American Mathematical Society