Nonintegrated 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), 11451162
MSC (2010):
Primary 32H30; Secondary 53A10
Published electronically:
October 24, 2011
MathSciNet review:
2869171
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, we establish a nonintegrated 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.
Ta
Thi Hoai An and Ha
Tran 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), no. 3, 775–786. MR 2534280
(2010i:30048)
 2.
Aldo
Biancofiore and Wilhelm
Stoll, Another proof of the lemma of the logarithmic derivative in
several complex variables, Recent developments in several complex
variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of
Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981,
pp. 29–45. MR 627748
(83i:32007)
 3.
Gerd
Dethloff and Tran
Van Tan, A second main theorem for moving hypersurface
targets, Houston J. Math. 37 (2011), no. 1,
79–111. MR
2786547 (2012g:32022)
 4.
Hirotaka
Fujimoto, On the Gauss map of a complete minimal surface in
𝑅^{𝑚}, J. Math. Soc. Japan 35 (1983),
no. 2, 279–288. MR 692327
(84g:53015), 10.2969/jmsj/03520279
 5.
Hirotaka
Fujimoto, Value distribution of the Gauss maps of complete minimal
surfaces in 𝑅^{𝑚}, J. Math. Soc. Japan
35 (1983), no. 4, 663–681. MR 714468
(85c:53011), 10.2969/jmsj/03540663
 6.
Hirotaka
Fujimoto, Nonintegrated defect relation for meromorphic maps of
complete Kähler manifolds into
𝑃^{𝑁₁}(𝐶)×\cdots×𝑃^{𝑁_{𝑘}}(𝐶),
Japan. J. Math. (N.S.) 11 (1985), no. 2,
233–264. MR
884636 (88m:32049)
 7.
W.
K. Hayman, Meromorphic functions, Oxford Mathematical
Monographs, Clarendon Press, Oxford, 1964. MR 0164038
(29 #1337)
 8.
Leon
Karp, Subharmonic functions on real and complex manifolds,
Math. Z. 179 (1982), no. 4, 535–554. MR 652859
(84d:53042), 10.1007/BF01215065
 9.
Yuancheng
Liu and Min
Ru, A defect relation for meromorphic maps on parabolic manifolds
intersecting hypersurfaces, Illinois J. Math. 49
(2005), no. 1, 237–257 (electronic). MR 2157377
(2006c:32014)
 10.
Min
Ru, A defect relation for holomorphic curves intersecting
hypersurfaces, Amer. J. Math. 126 (2004), no. 1,
215–226. MR 2033568
(2004k:32026)
 11.
Min
Ru, Holomorphic curves into algebraic varieties, Ann. of Math.
(2) 169 (2009), no. 1, 255–267. MR 2480605
(2010d:32009), 10.4007/annals.2009.169.255
 12.
Wilhelm
Stoll, Introduction to value distribution theory of meromorphic
maps, Complex analysis (Trieste, 1980) Lecture Notes in Math.,
vol. 950, Springer, BerlinNew York, 1982, pp. 210–359. MR 672787
(84a:32041)
 13.
Shing
Tung Yau, Some functiontheoretic properties of complete Riemannian
manifold and their applications to geometry, Indiana Univ. Math. J.
25 (1976), no. 7, 659–670. MR 0417452
(54 #5502)
 14.
Al
Vitter, The lemma of the logarithmic derivative in several complex
variables, Duke Math. J. 44 (1977), no. 1,
89–104. MR
0432924 (55 #5903)
 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), 775786. 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), 2945. MR 627748 (83i:32007)
 3.
 G. Dethloff and T. V. Tan, A second main theorem for moving hypersurfaces targets, Houston J. Math., 37 (2011), 79111. MR 2786547
 4.
 H. Fujimoto, On the Gauss map of a complete minimal surface in , J. Math. Soc. Japan, 35 (1983), No.2, 279288. MR 692327 (84g:53015)
 5.
 H. Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in , J. Math. Soc. Japan, 35 (1983), 663681. 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, 233264. 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), 535554. 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, 237257. MR 2157377 (2006c:32014)
 10.
 M. Ru, A defect relation for holomorphic curves intersecting hypersurfaces, American Journal of Mathematics 126 (2004), 215226. MR 2033568 (2004k:32026)
 11.
 M. Ru, Holomorphic curves into algebraic varieties, Ann. of Math. (2), 169 (2009), no.1, 255267. MR 2480605 (2010d:32009)
 12.
 W. Stoll, Introduction to value distribution theory of meromorphic maps, Lecture Notes in Math. 950 (1982), 210359. MR 672787 (84a:32041)
 13.
 S.T. Yau, Some functiontheoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana U. Math. J. 25 (1976), 659670. MR 0417452 (54:5502)
 14.
 A. Vitter, The lemma of the logarithmic derivative in several variables, Duke Math. J., 44(1977), 89104. MR 0432924 (55:5903)
Similar Articles
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/S000299472011055121
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 H982300910004 and H98230110201
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
