Abstract
In this article, we give a survey of recent development of uniqueness problem for meromorphic mappings. In particular, we give an overview of its applications to constructing problem of hyperbolic hypersurfaces in complex projective spaces. Furthermore, we give a review on some recent researches on unique range set for meromorphic functions of one complex variable.
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References
Aihara Y., A unicity theorem for meromorphic mappings into compactified locally symmetric spaces, Kodai Math. J. 14 (1991), 392–405.
Aihara Y., Finiteness theorems for meromorphic mappings, Osaka J. Math. 35 (1998), 593–616.
Aihara Y., The uniqueness problem of meromorphic mappings with deficiencies, Tohoku Math. J. 51 (1999), 315–328.
Aihara Y., Unicity theorems for meromorphic mappings with deficiencies, Complex Variables Theory Appl. 42 (2000).
Aihara Y., Propagation of algebraic dependence of meromorphic mappings, Taiwanese J. Math. 5 (2001), 667–679.
Aihara Y., Algebraic dependence of meromorphic mappings in value distribution theory, to appear in Nagoya Math. J. 169 (2003).
Cartan H., Sur les systèmes de fonctions holomorphes à variété linéaires lacunaires et leurs applications, Ann. Sci. École Norm. Sup. 45 (1928), 255–346.
Cartan H., Sur les systèmes de fonctions holomorphes à variété linéaires lacunaires et leurs applications, C. R. École Norm. Sup. 45 (1928), 255–346.
Drouilhet S. J., A unicity theorem for meromorphic mappings between algebraic varieties, Trans. Amer. Math. Soc. 265 (1981), 349–358.
Drouilhet S. J., Criteria for algebraic dependence of meromorphic mappings into algebraic varieties, Illinois J. Math. 26 (1982), 492–502.
Fujimoto H., On meromorphic maps into the complex projective space, J. Math. Soc. Japan, 26 (1974), 271–288.
Fujimoto H., The uniqueness problem of meromorphic maps into the complex projective spaces, Nagoya Math. J. 58 (1975), 1–23.
Fujimoto H., A uniqueness theorem for algebraically non-degenerate meromorphic maps into PN(ℂ), Nagoya Math. J. 64 (1976), 117–147.
Fujimoto H., Remarks to the uniqueness problems of meromorphic mappings into PN(ℂ), I–IV, Nagoya Math. J. 71 (1978), 13–24, 25–41; ibid. 75 (1979), 71–85; ibid. 83 (1981), 153–181.
Fujimoto H., On meromorphic maps into compact complex manifolds, J. Math. Soc. Japan 34 (1982), 527–539.
Fujimoto H., Finiteness of some families of meromorphic maps, Kodai Math. J. 11 (1988), 47–63.
Fujimoto H., Uniqueness problem with truncated multiplicities in value distribution theory I, II, Nagoya Math. J. 152 (1998), 131–152; ibid. 155 (1999), 161–188.
Fujimoto H., Uniqueness problem with truncated multiplicities in value distribution theory I, II, Nagoya Math. J. 152 (1998), 131–152; ibid. 155 (1999), 161–188.
Fujimoto H., On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122 (2000), 1175–1203.
Fujimoto H., Uniqueness polynomials, to appear in Nagoya Math. J. 170 (2003).
Green M., Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43–75.
Gross F. and Yang C.C., On preimage and range sets of meromorphic functions, Proc. Japan Acad. 58 (1982), 17–20.
Ji S., Uniqueness problem without multiplicities in value distribution theory, Pacific J. Math. 135 (1988), 223–248.
Lang S., Higher dimensional Diophantine problem, Bull. Amer. Math. Soc. 80 (1974), 779–787.
Lang S., Hyperbolic and Diophantine analysis, Amer. Math. Soc. 14 (1986), 159–205.
Lang S., Introduction to Complex Hyperbolic Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1987.
Li P. and Yang C.C., Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), 437–450.
Mausda K. and Noguchi J., A construction of hyperbolic hypersurfaces of P n (C), Math. Ann. 304 (1996), 339–362.
Nevanlinna R., Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen, Acta Math. 48 (1926), 367–391.
Nevanlinna R., Compléments aux théorèmes d’unicité dans la théorie des fonctions méromorphes, Comptes Rendus 186 (1928), 289–291.
Nevanlinna R., Le Théorème de Picard-Borel et la Théorie des Fonction Méromorphes, Gauthier-Villars, Paris, 1929.
Noguchi J., Meromorphic mappings of covering spaces over ℂminto a projective variety and defect relations, Hiroshima Math. J. 6 (1976), 265–280.
Noguchi J., Holomorphic mappings into closed Riemann surfaces, Hiroshima Math. J. 6 (1976), 281–290.
Noguchi J., Hyperbolic fiber spaces and Mordell’s conjecture over function fields, Publ. Res. Inst. Math. Sci. 21 (1985), 27–46.
Noguchi J., Meromorphic mappings into compact hyperbolic complex spaces and geometric Diophantine problem, Internat. J. Math. 3 (1992), 277–289.
Noguchi J., An arithmetic property of Shirosaki’s hyperbolic projective hypersurface, to appear in Forum Math.
Noguchi J. and Ochiai T., Geometric Function Theory in Several Complex Variables, Trans. Math. Monographs 80, Amer. Math. Soc. (1990).
Pólya G., Bestimmung einer ganzen Funktionen endlichen Geschlechts durch vierelei Stellen, Math. Tidsskrift B (1921), 16–21.
Sario L., General value distribution theory, Nagoya Math. J. 23 (1963), 213–229.
Schmid E. M., Some theorems on value distribution of meromorphic functions, Math. Z. 120 (1971), 61–92.
Selberg H., Algebroid Funktionen und Umkerfunktionen Abelscher Integrale, Avh. Norske Vid. Acad. Oslo 8 (1934), 1–72.
Shiffman B., Uniqueness of entire and meromorphic functions sharing finite sets, Complex Variables Theory Appl. 43 (2001), 433–449.
Shirosaki M., On polynomials which determines holomorphic mappings, J. Math. Soc. Japan 49 (1997), 289–298.
Shirosaki M. On some hypersurfaces and holomorphic mappings, Kodai Math. J. 21 (1998), 29–34.
Shirosaki M., A hypersurface which determines linearly nondegenerate holomorphic mappings, Kodai Math. J. 23 (2000), 105–107.
Shirosaki M., Hyperbolic hypersurface in the complex projective spaces of low dimensions, Kodai Math. J. 23 (2000), 243–233.
Silverman J., The Arithmetic of Elliptic Curves, Springer-Verlag, Berlin-Heidelberg-New York, 1991.
Smiley L., Dependence theorems for meromorphic maps, Ph.D. Thesis, Notre Dame Univ., 1979.
Smiley L., Geometric conditions for the unicity of holomorphic curves, Contemporary Math. 25 (1983), 149–154.
Stoll W., The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Proc. Value Distribution Theory, Joensuu 1981 (eds. I. Laine et al.), pp. 101–219, Lect. Notes in Math. 981, Springer-Verlag, Berlin-Heidelberg-New York, 1983.
Stoll W., Algebroid reduction of Nevanlinna theory, Complex Analysis III, Proc. 1985–1986, (ed. C. A. Berenstein), pp. 131–241, Lect. Notes in Math. 1277, Springer-Verlag, Berlin-Heidelberg-New York, 1987.
Stoll W., On the propagation of dependences, Pacific J. Math. 139 (1989), 311–337.
Yang C. C. and Hua X., Unique polynomials of entire and meromorphic functions, Mat. Fiz. Anal. Geom. 4 (1997), 391–398.
Yi H.-X., A question of Gross and the uniqueness of entire functions, Nagoya Math. J. 138 (1995), 169–177.
Yi H.-X., The unique range sets of entire or meromorphic functions, Complex Variables Theory Appl. 28 (1995), 13–21.
Yi H.-X., Some further results on uniqueness of meromorphic functions, Complex Variables Theory Appl. 38 (1999), 375–385.
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Aihara, Y. (2004). Recent Topics in Uniqueness Problem for Meromorphic Mappings. In: Barsegian, G., Laine, I., Yang, C.C. (eds) Value Distribution Theory and Related Topics. Advances in Complex Analysis and Its Applications, vol 3. Springer, Boston, MA. https://doi.org/10.1007/1-4020-7951-6_13
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DOI: https://doi.org/10.1007/1-4020-7951-6_13
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