Skip to main content
SpringerLink
Log in

The ahlfors — Weyl theory of meromorphic maps on parabolic manifolds

  • Conference paper
  • First Online:
Value Distribution Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 981))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

16. References

  1. Ahlfors, L. V.: The theory of meromorphic curves. Acta Soc. Sci. Fenn. Nova Ser A 3 (4) (1941), 1–31.

    MathSciNet  MATH  Google Scholar 

  2. Biancofiore, A.: A hypersurface defect relation for a class of meromorphic maps. Trans. Amer. Math. Soc. 270 (1982), 47–60.

    Article  MathSciNet  MATH  Google Scholar 

  3. Biancofiore, A.: A defect relation for meromorphic maps. Manuscript.

    Google Scholar 

  4. Biancofiore, A.: A defect relation for linear systems on compact complex monifolds. Manuscript.

    Google Scholar 

  5. Biancofiore, A., Stoll W.: Another proof of the lemma of the logarithmic derivative in several complex variables, in "Recent developments in several complex variables". Annals of Math. Studies 100, Princeton University Press, Princeton, N. J. (1981), 29–45.

    Google Scholar 

  6. Borel, E.: Sur les zéros des fonctions entières. Acta Math. 20 (1897), 357–396.

    Article  MathSciNet  Google Scholar 

  7. Bott, R., Chern, S. S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114 (1965), 71–112.

    Article  MathSciNet  MATH  Google Scholar 

  8. Burns, D.: Curvature of Monge-Ampère foliations and parabolic manifolds. Manuscript.

    Google Scholar 

  9. Carlson J., Griffiths Ph.: A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. of Math. 95 (1972), 557–584.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cartan, H.: Un noveau théoreme d'unicité relatif aux fonctions méromorphes. C.R. Acad. Sci. 188 (1929), 301–303.

    MATH  Google Scholar 

  11. Cartan, H.: Sur les zéros des combinaisons linéaires des fonctions holomorphes données. Mathematica (Clnj) 7 (1933), 5–31.

    MathSciNet  MATH  Google Scholar 

  12. Chern, S. S.: The integrated form of the first main theorem for complex analytic mappings in several variables, Ann. of Math. 77 (1960), 536–551.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chern, S. S.: Holomorphic curves in the plane, in "Diff. Geom. in honor of K. Yano". Kinokuniya, Tokyo (1972), 72–94.

    Google Scholar 

  14. Cowen, M.: Hermitian vector bundles and value distribution for Schubert cycles. Trans. Amer. Math. Soc. 180 (1973), 189–228.

    Article  MathSciNet  MATH  Google Scholar 

  15. Cowen, M., Griffiths, Ph.: Holomorphic curves and metrics of negative curvature, J. Analyse Math. 29 (1976), 93–152.

    Article  MathSciNet  MATH  Google Scholar 

  16. Fujimoto, H.: The uniqueness problem of meromorphic maps into the complex projective space. Nagoya Math. J. 58 (1975), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujimoto, H.: The uniqueness theorem of algebraically non-degenerate meromorphic maps into IPn (ℂ). Nagoya Math. J. 64 (1976), 117–147.

    Article  MathSciNet  Google Scholar 

  18. Fujimoto, H.: Remarks to the uniqueness problem of meromorphic maps I, III. Nagoya Math. J. 71 (1978) 13–41, 75 (1979) 71–85.

    Article  MathSciNet  MATH  Google Scholar 

  19. Fujimoto, H.: On the defect relation of the derived curves of a holomorphic curve in IPn (ℂ). Tôhoku Math. J. 34 (1982), 141–160.

    Article  MathSciNet  Google Scholar 

  20. Green, M.: Holomorphic maps into complex projective space omitting hyperplanes. Trans. Amer. Math. Soc. 169 (1972), 89–103.

    Article  MathSciNet  MATH  Google Scholar 

  21. Green, M.: Some Picard theorems for holomorphic maps to algebraic varieties. Amer. J. Math. 97 (1975), 43–75.

    Article  MathSciNet  MATH  Google Scholar 

  22. Green, M.: Some examples and counter-examples in value distribution theory for several variables. Compositio Math. 30 (1975), 317–322.

    MathSciNet  MATH  Google Scholar 

  23. Griffiths, Ph.: Holomorphic mappings into canonical algebraic varieties. Ann. of Math. 93 (1971), 439–458.

    Article  MathSciNet  MATH  Google Scholar 

  24. Griffiths, Ph.: Holomorphic mappings: Survey of some results and discussion of open problems. Bull. Amer. Math. Soc. 78 (1972), 374–382.

    Article  MathSciNet  MATH  Google Scholar 

  25. Griffiths, Ph., King, J.: Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220.

    Article  MathSciNet  MATH  Google Scholar 

  26. Griffiths, Ph.: Entire holomorphic mappings in one and several complex variables. Annals of Math. Studies 85, Princeton University Press, Princeton, N. J. (1976).

    Book  MATH  Google Scholar 

  27. Gruman, L.: Value distribution for holomorphic maps into ℂn. Math. Ann. 245 (1979), 29–37.

    Article  MathSciNet  Google Scholar 

  28. Gruman, L.: La geometrie globale des ensembles analytiques dans ℂn, in "Séminaire P. Lelong-H. Skoda (Analyse). Années 1978/79". Lecture Notes in Mathematics 822, Springer-Verlag, Berlin-Heidelberg-New York (1980), 90–98.

    Chapter  Google Scholar 

  29. Hayman, W. K.: Meromorphic functions. Oxford University Press, Oxford (1964).

    MATH  Google Scholar 

  30. Hirschfelder, J.: The first main theorem of value distribution in several variables. Invent. Math. 8 (1969), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kataka Conference: Open problems in geometric function theory. Proceed. 5. Int. Symp. Division Math. The T niguchi Foundation (available from: Dept. of Math. Osaka Univ., Toyonaka, 560 Japan) (1978).

    Google Scholar 

  32. Kneser, H.: Zur Theorie der gebrochenen Funktionen mehrerer Veränderlichen. Jahresber. Deutsch. Math.-Verein. 48 (1938), 1–28.

    MathSciNet  MATH  Google Scholar 

  33. Lelong, P.: Fonctions entiéres (n-variables) et fonctions plurisous-harmoniques d'ordre fini dans ℂn. J. Analyse Math. 12 (1964), 365–407.

    Article  MathSciNet  MATH  Google Scholar 

  34. Levine, H.: A theorem on holomorphic mappings into complex projective space. Ann. of Math. 71 (1960), 529–535.

    Article  MathSciNet  MATH  Google Scholar 

  35. Molzon, R. E.: Sets omitted by equidimensional holomorphic mappings Amer. J. Math. 101 (1979), 1271–1283.

    Article  MathSciNet  MATH  Google Scholar 

  36. Molzon, R. E., Shiffman, B., Sibony, N.: Average growth estimates for hyperplane sections of entire analytic sets. Math. Ann. 257 (1981), 43–59.

    Article  MathSciNet  MATH  Google Scholar 

  37. Mori, S.: On the deficiencies of meromorphic maps of ℂm into INN(ℂ). Nagoya Math. J. 67 (1977), 165–176.

    Article  MathSciNet  Google Scholar 

  38. Mori, S.: The deficiencies and the order of holomorphic mappings of ℂn into a compact complex manifold. Tôhoku Math. J. 31 (1979), 285–291.

    Article  MathSciNet  MATH  Google Scholar 

  39. Mori, S.: Holomorphic curves with maximal deficiency sum. Kodai Math. J. 2 (1979), 116–122.

    Article  MathSciNet  MATH  Google Scholar 

  40. Murray, J.: A second main theorem of value distribution theory on Stein manifolds with pseudoconvex exhaustion (1974). Notre Dame Thesis.

    Google Scholar 

  41. Nevanlinna, R.: Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen. Acta Math. 48 (1926), 367–391.

    Article  MathSciNet  MATH  Google Scholar 

  42. Nevanlinna, R.: Eindeutige analytische Funktionen. 2. Auflage. Springer-Verlag, Berlin-Göttingen-Heidelberg (1953).

    Book  MATH  Google Scholar 

  43. Noguchi, J.: A relation between order and defects of meromorphic mappings of ℂn into IPN(ℂ). Nagoya Math. J. 59 (1975), 97–106.

    Article  MathSciNet  Google Scholar 

  44. Noguchi, J.: Holomorphic curves in algebraic varieties. Hiroshima Math. J. 6 (1977), 833–853. Supplement: Hiroshima Math. J. 10 (1980), 229–231.

    MathSciNet  MATH  Google Scholar 

  45. Noguchi, J., Sunada, T.: Finiteness of the family of rational and meromorphic mappings into algebraic varieties. Manuscript.

    Google Scholar 

  46. Ronkin, L. I.: Introduction to the theory of entire functions of several variables. Translations of Mathematical Monographs 44, Amer. Math. Soc., Providence, R. I. (1974).

    MATH  Google Scholar 

  47. Sadullaev, A.: Deficient divisors in the Valiron sense. Math USSR-Sb. 36 (1980), 535–547.

    Article  MATH  Google Scholar 

  48. Sakai, F.: Degeneracy of holomorphic maps with ramification. Invent. Math. 26 (1974), 213–229.

    Article  MathSciNet  MATH  Google Scholar 

  49. Shiffman, B.: Holomorphic and meromorphic mappings and curvature. Math. Ann. 222 (1976), 171–194.

    Article  MathSciNet  MATH  Google Scholar 

  50. Shiffman, B.: Applications of geometric measure theory to value distribution theory for meromorphic maps, in "Value-Distribution Theory Part A", Marcel Dekker, Inc., New York (1974), 63–95.

    Google Scholar 

  51. Shiffman, B.: Nevanlinna defect relations for singular divisors. Invent. Math. 31 (1975), 155–182.

    Article  MathSciNet  MATH  Google Scholar 

  52. Shiffman, B.: Holomorphic curves in algebraic manifolds. Bull. Amer. Math. Soc. 83 (1977), 155–182.

    Article  MathSciNet  MATH  Google Scholar 

  53. Shiffman, B.: On holomorphic curves and meromorphic maps in projective space. Indiana Univ. Math. J. 28 (1979), 627–641.

    Article  MathSciNet  MATH  Google Scholar 

  54. Skoda, H.: Solution a croissance du second problème de Cousin dans ℂn. Ann. Inst. Fourier (Grenoble) 21 (1971), 11–23.

    Article  MathSciNet  MATH  Google Scholar 

  55. Skoda, H.: Sous-ensembles analytiques d'ordre fini ou infini dans ℂn. Bull. Soc. Math. France 100 (1972), 353–408.

    MathSciNet  MATH  Google Scholar 

  56. Smiley, L.: Dependence theorems for meromorphic maps. (1979). Notre Dame Thesis.

    Google Scholar 

  57. Stoll, W.: Mehrfache Integrale auf komplexen Mannigfaltigkeiten. Math. Z. 57 (1952), 116–154.

    Article  MathSciNet  MATH  Google Scholar 

  58. Stoll, W.: Ganze Funktionen endlicher Ordnung mit gegebenen Nullstellenflächen. Math. Z. 57 (1953), 211–237.

    Article  MathSciNet  MATH  Google Scholar 

  59. Stoll, W.: Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen I, II. Acta Math. 90 (1953), 1–115, 92 (1954), 55–169.

    Article  MathSciNet  MATH  Google Scholar 

  60. Stoll, W.: The growth of the area of a transcendental analytic set I, II Math. Ann. 156 (1964), 47–78, 156 (1964), 144–170.

    Article  MathSciNet  MATH  Google Scholar 

  61. Stoll, W.: The multiplicity of a holomorphic map. Invent. Math. 2 (1966), 154–218.

    Article  MathSciNet  MATH  Google Scholar 

  62. Stoll, W.: A general first main theorem of value distribution. Acta Math. 118 (1967), 111–191.

    Article  MathSciNet  MATH  Google Scholar 

  63. Stoll, W.: About value distribution of holomorphic maps into projective space. Acta Math. 123 (1969), 83–114.

    Article  MathSciNet  MATH  Google Scholar 

  64. Stoll, W.: Value distribution of holovorphic maps into compact complex manifolds. Lecture Notes in Mathematics 135, Springer-Verlag, Berlin-Heidelberg-New York (1970).

    Book  MATH  Google Scholar 

  65. Stoll, W.: Value distribution of holomorphic maps, in "Several complex variables I, Maryland 1970". Lecture Notes in Mathematics 155. Springer-Verlag, Berlin-Heidelberg-New York (1970), 165–170.

    Chapter  Google Scholar 

  66. Stoll, W.: Fiber integration and some of its applications, in "Symposium on Several Complex Variables, Park City, Utah, 1970". Lecture Notes in Mathematics 184, Springer-Verlag, Berlin-Heidelberg-New York (1971), 109–120.

    Chapter  Google Scholar 

  67. Stoll, W.: Deficit and Bezout estimates, in "Value — Distribution Theory Part B". Marcel Dekker, Inc., New York (1973).

    Google Scholar 

  68. Stoll, W.: Holomorphic functions of finite order in several complex variables. CBMS. Regional Conference Series in Mathematics 21, Amer. Math. Soc., Providence, R. I. (1974).

    MATH  Google Scholar 

  69. Stoll, W.: Aspects of value distribution theory in several complex variables. Bull. Amer. Math. Soc. 83 (1977), 166–183.

    Article  MathSciNet  MATH  Google Scholar 

  70. Stoll, W.: Value distribution theory on parabolic spaces. Lecture Notes in Mathematics 600, Springer-Verlag, Berlin-Heidelberg-New York (1977).

    Book  MATH  Google Scholar 

  71. Stoll, W.: A Casorati — Weierstrass Theorem for Schubert zeroes in semi-ample holomorphic vector bundles. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 15 (1978), 63–90.

    MathSciNet  MATH  Google Scholar 

  72. Stoll, W.: The characterization of strictly parabolic manifolds. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) 7 (1980), 87–154.

    MathSciNet  MATH  Google Scholar 

  73. Stoll, W.: The characterization of strictly parabolic spaces. Compositio Math. 44 (1981), 305–373.

    MathSciNet  MATH  Google Scholar 

  74. Stoll, W.: Introduction to value distribution theory of meromorphic maps. To appear in Proceeding of the 1980 Trieste Conference, Lecture Notes in Mathematics, Springer-Verlag.

    Google Scholar 

  75. Sung, C. H.: On the deficiencies of holomorphic curves in projective space. Manuscript.

    Google Scholar 

  76. Tung, Ch.: The first main theorem of value distribution on complex spaces. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 15 (1979), 91–263.

    MathSciNet  Google Scholar 

  77. Vitter, A.: The lemma of the logarithmic derivative in several complex variables. Duke Math. J. 44 (1977), 89–104.

    Article  MathSciNet  MATH  Google Scholar 

  78. Weyl, H., Weyl, J.: Meromorphic curves. Ann. of Math. 39 (1938), 516–538.

    Article  MathSciNet  MATH  Google Scholar 

  79. Weyl, H., Weyl, J.: Meromorphic functions and analytic curves. Annals of Math. Studies 12, Princeton University Press, Princeton, N. J. (1943).

    MATH  Google Scholar 

  80. Wirtinger, W.: Ein Integralsatz über analytische Gebilde im Gebiete von mehreren komplexen Veränderlichen. Monatshefte Math. Phys. 45 (1937), 418–431.

    Article  MathSciNet  MATH  Google Scholar 

  81. Wong, P.-M.: Defect relations for maps on parabolic spaces and Kobayashi metrics on projective spaces omitting hyperplanes. (1976) Notre Dame Thesis.

    Google Scholar 

  82. Wong, P.-M.: Geometry of the complex homogeneous Monge — Ampére equation. Invent. Math. 67 (1982), 261–274.

    Article  MathSciNet  MATH  Google Scholar 

  83. Wu, H.: Mappings of Riemann surfaces (Nevanlinna Theory). Proc. Sympos. Pure Math. 11 (1968), 480–552.

    Article  MathSciNet  Google Scholar 

  84. Wu, H.: Remarks on the first main teorem of equidistribution theory I–IV. J. Differential Geom. 2 (1968), 197–202 and 369–384, 3 (1969), 83–94 and 433–446.

    Google Scholar 

  85. Wu, H.: The equidistribution theory of holomorphic curves. Annals of Math. Studies 64, Princeton University Press, Princeton, N.J. (1970).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 46556, Notre Dame, IN, U.S.A.

    Wilhelm Stoll

Authors
  1. Wilhelm Stoll
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ilpo Laine Seppo Rickman

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Stoll, W. (1983). The ahlfors — Weyl theory of meromorphic maps on parabolic manifolds. In: Laine, I., Rickman, S. (eds) Value Distribution Theory. Lecture Notes in Mathematics, vol 981. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066385

Download citation

  • DOI: https://doi.org/10.1007/BFb0066385

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12003-2

  • Online ISBN: 978-3-540-39480-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation