Advertisement

Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 211–229 | Cite as

Non-integrated defect of meromorphic maps on Kähler manifolds

  • Do Duc ThaiEmail author
  • Si Duc Quang
Article
  • 95 Downloads

Abstract

The purpose of this article is twofold. The first is to establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold quotien of a ball into a projective variety intersecting hypersurfaces in subgeneral position. We also apply it to the Gauss mapping from a closed regular submanifold of \({\mathbb {C}}^m\). The second aim is to establish an above type theorem with truncation level 1 for differentiably nondegenerate meromorphic mappings.

Keywords

Nevanlinna theory Second main theorem Meromorphic mapping Non-integrated defect relation 

Mathematics Subject Classification

Primary 32H30 32A22 Secondary 30D35 

References

  1. 1.
    Fujimoto, H.: On the Gauss mapping from a complete minimal surface in \({\mathbf{R}}^m\). J. Math. Soc. Jpn. 35, 279–288 (1983)CrossRefGoogle Scholar
  2. 2.
    Fujimoto, H.: Value distribution of the Gauss mappings from complete minimal surfaces in \({\mathbf{R}}^m\). J. Math. Soc. Jpn. 35, 663–681 (1983)CrossRefzbMATHGoogle Scholar
  3. 3.
    Fujimoto, H.: Non-integrated defect relation for meromorphic mappings from complete Kähler manifolds into \({\mathbb{P}}^{N_1}(\mathbb{C})\times \cdots \times {\mathbb{P}}^{N_k}(\mathbb{C})\). Jpn. J. Math. 11, 233–264 (1985)CrossRefGoogle Scholar
  4. 4.
    Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179, 535–554 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Noguchi, J.: A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka’s theorem. Kodai Math. J. 28, 336–346 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Noguchi, J., Winkelmann, J.: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation, Grundlehren der mathematischen Wissenschaften 350. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Quang, S.D., An, D.P.: Second main theorem and unicity of meromorphic mappings for hypersurfaces in projective varieties. Acta Math. Vietnam. 42, 455–470 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Quang, S.D., Phuong, N.T.Q., Nhung, N.T.: Non-integrated defect relation for meromorphic maps from a Kahler manifold intersecting hypersurfaces in subgeneral of position. J. Math. Anal. Appl. 452, 1434–1452 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ru, M., Sogome, S.: Non-integrated defect relation for meromorphic mappings from complete Kähler manifolds into \({\mathbb{P}}^n({\mathbb{C}})\) intersecting hypersurfaces. Trans. Am. Math. Soc. 364, 1145–1162 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Tan, T.V., Truong, V.V.: A non-integrated defect relation for meromorphic mappings from complete Kähler manifolds into a projective variety intersecting hypersurfaces. Bull. Sci. Math. 136, 111–126 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Thai, D.D., Viet, V.D.: Holomorphic mappings into compact complex manifolds. Houston J. Math. 43(3), 725–762 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yan, Q.: Non-integrated defect relation and uniqueness theorem for meromorphic mappings from a complete Kähler manifold into \(P^n(\mathbb{C})\). J. Math. Anal. Appl. 398, 567–581 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yau, S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana U. Math. J. 25, 659–670 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Thang Long Instutute of Mathematics and Applied SciencesHanoiVietnam

Personalised recommendations