Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into intersecting hypersurfaces

Authors:
Min Ru and Suraizou Sogome

Journal:
Trans. Amer. Math. Soc. **364** (2012), 1145-1162

MSC (2010):
Primary 32H30; Secondary 53A10

Published electronically:
October 24, 2011

MathSciNet review:
2869171

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we establish a non-integrated defect relation for a meromorphic map of a complete Kähler manifold whose universal covering is biholomorphic to a ball in into intersecting hypersurfaces in general position, as well as an application to the Gauss map of a closed regular submanifold of . The result provides a complement to the recent result of Ru (2004) on a defect relation for meromorphic mappings from into intersecting hypersurfaces in general position.

**1.**T. T. H. An and H. T. Phuong, An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space,*Houston J. Math.*,**35**(2009), 775-786. MR**2534280 (2010i:30048)****2.**A. Biancofiore and W. Stoll, Another proof of the lemma of the logarithmic derivative in several complex variables,*Ann. of Math. Studies*,**100**(1981), 29-45. MR**627748 (83i:32007)****3.**G. Dethloff and T. V. Tan, A second main theorem for moving hypersurfaces targets,*Houston J. Math.*,**37**(2011), 79-111. MR**2786547****4.**H. Fujimoto, On the Gauss map of a complete minimal surface in ,*J. Math. Soc. Japan*,**35**(1983), No.2, 279-288. MR**692327 (84g:53015)****5.**H. Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in ,*J. Math. Soc. Japan*,**35**(1983), 663-681. MR**714468 (85c:53011)****6.**H. Fujimoto, Nonintegrated defect relation for meromorphic maps of complete Kähler manifolds into ,*Japan. J. Math.*,**11**(1985), No 2, 233-264. MR**884636 (88m:32049)****7.**W.K. Hayman, Meromorphic functions,*Oxford Mathematical Monographs,*Clarendon Press, 1964. MR**0164038 (29:1337)****8.**L. Karp, Subharmonic functions on real and complex manifolds,*Math. Z.*,**179**(1982), 535-554. MR**652859 (84d:53042)****9.**Y. Liu and M. Ru, A defect relation for meromorphic maps on parabolic manifolds intersecting hypersurfaces,*Illinois J. Math.*,**49**(2005), no. 1, 237-257. MR**2157377 (2006c:32014)****10.**M. Ru, A defect relation for holomorphic curves intersecting hypersurfaces,*American Journal of Mathematics***126**(2004), 215-226. MR**2033568 (2004k:32026)****11.**M. Ru, Holomorphic curves into algebraic varieties,*Ann. of Math. (2)*,**169**(2009), no.1, 255-267. MR**2480605 (2010d:32009)****12.**W. Stoll, Introduction to value distribution theory of meromorphic maps,*Lecture Notes in Math.***950**(1982), 210-359. MR**672787 (84a:32041)****13.**S.-T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry,*Indiana U. Math. J.***25**(1976), 659-670. MR**0417452 (54:5502)****14.**A. Vitter, The lemma of the logarithmic derivative in several variables,*Duke Math. J.*, 44(1977), 89-104. MR**0432924 (55:5903)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
32H30,
53A10

Retrieve articles in all journals with MSC (2010): 32H30, 53A10

Additional Information

**Min Ru**

Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204

Email:
minru@math.uh.edu

**Suraizou Sogome**

Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204

Email:
ssogome@math.uh.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05512-1

Keywords:
Meromorphic mappings,
Gauss map of minimal surfaces,
defect relation,
Nevanlinna theory

Received by editor(s):
January 26, 2010

Published electronically:
October 24, 2011

Additional Notes:
The first author was supported in part by NSA under grant number H98230-09-1-0004 and H98230-11-0201

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.