A unicity theorem for meromorphic maps of a complete Kähler manifold into ℙⁿ(ℂ) sharing hypersurfaces

Ru, Min; Sogome, Suraizou

doi:10.1090/S0002-9939-2013-11718-1

A unicity theorem for meromorphic maps of a complete Kähler manifold into $\mathbb {P}^n(\mathbb {C})$ sharing hypersurfaces
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by Min Ru and Suraizou Sogome PDF

Proc. Amer. Math. Soc. 141 (2013), 4229-4239 Request permission

Abstract:

In this paper, we give a unicity theorem for meromorphic maps of an $m-$dimensional complete Kähler manifold $M$, whose universal covering is a ball in $\mathbb {C}^m$, into $\mathbb {P}^n(\mathbb {C})$, sharing the hypersurfaces in general position in $\mathbb {P}^n(\mathbb {C})$, where the maps satisfy a certain growth condition.

References

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
Hirotaka Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory, Nagoya Math. J. 152 (1998), 131–152. MR 1659377, DOI 10.1017/S0027763000006826
Hirotaka Fujimoto, Nonintegrated defect relation for meromorphic maps of complete Kähler manifolds into $P^{N_1}(\textbf {C})\times \cdots \times P^{N_k}(\textbf {C})$, Japan. J. Math. (N.S.) 11 (1985), no. 2, 233–264. MR 884636, DOI 10.4099/math1924.11.233
Hirotaka Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in $\textbf {R}^{m}$, J. Math. Soc. Japan 35 (1983), no. 4, 663–681. MR 714468, DOI 10.2969/jmsj/03540663
Hirotaka Fujimoto, A unicity theorem for meromorphic maps of a complete Kähler manifold into $P^N(\textbf {C})$, Tohoku Math. J. (2) 38 (1986), no. 2, 327–341. MR 843816, DOI 10.2748/tmj/1178228497
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
Leon Karp, Subharmonic functions on real and complex manifolds, Math. Z. 179 (1982), no. 4, 535–554. MR 652859, DOI 10.1007/BF01215065
Rolf Nevanlinna, Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen, Acta Math. 48 (1926), no. 3-4, 367–391 (German). MR 1555233, DOI 10.1007/BF02565342
Min Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), no. 1, 215–226. MR 2033568, DOI 10.1353/ajm.2004.0006
Min Ru, Holomorphic curves into algebraic varieties, Ann. of Math. (2) 169 (2009), no. 1, 255–267. MR 2480605, DOI 10.4007/annals.2009.169.255
Min Ru and Suraizou Sogome, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into $\Bbb {P}^{n}(\Bbb {C})$ intersecting hypersurfaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1145–1162. MR 2869171, DOI 10.1090/S0002-9947-2011-05512-1
Matthew Dulock and Min Ru, A uniqueness theorem for holomorphic curves sharing hypersurfaces, Complex Var. Elliptic Equ. 53 (2008), no. 8, 797–802. MR 2436255, DOI 10.1080/17476930802127081
Wilhelm Stoll, Introduction to value distribution theory of meromorphic maps, Complex analysis (Trieste, 1980) Lecture Notes in Math., vol. 950, Springer, Berlin-New York, 1982, pp. 210–359. MR 672787
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051
Al Vitter, The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), no. 1, 89–104. MR 432924, DOI 10.1215/S0012-7094-77-04404-0