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Uniqueness theorems for holomorphic curves on annulus sharing hyperplanes

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In this paper, by using the second main theorem for holomorphic curves from annuli ∆ to P n(C) intersecting a collection of fixed hyperplanes in general position with truncated counting functions, we will prove some theorems on unicity for linearly non-degenerate holomorphic curves on annulus ignoring multiplicity with hyperplanes in general position in projective space. This theorems have shown the sufficient conditions for two linearly non-degenerate holomorphic curves being equivalent. | UNIQUENESS THEOREMS FOR HOLOMORPHIC CURVES ON ANNULUS SHARING HYPERPLANES Nguyen Viet Phuong Thai Nguyen University of Economics and Business Administration - Thai Nguyen University ABSTRACT In this paper, by using the second main theorem for holomorphic curves from annuli ∆ to Pn (C) inter- secting a collection of fixed hyperplanes in general position with truncated counting functions, we will prove some theorems on unicity for linearly non-degenerate holomorphic curves on annulus ignoring multiplicity with hyperplanes in general position in projective space. This theorems have shown the sufficient conditions for two linearly non-degenerate holomorphic curves being equivalent. Keywords: 1 Unicity, annuli, hyperplane, holomorphic curve, general position. INTRODUCTION In 1926, R. Nevanlinna proved that two nonconstant meromorphic functions of one complex variable which attain same five distinct values at the same points, must be identical. In 1975, H.Fujimoto (see [2]) generalized Nevanlinna’s result to the case of meromorphic mappings of Cm into Pn (C). He given the sufficient condition with 3n + 2 hyperplanes in general position which determining a meromorphic maps. Since that time, this problem has been studied intensively. The many mathematicians study two following problems: finding properties of unique range sets, and finding out a unique range set with the smallest number of elements as possible. For example: Fujimoto ([2],[3]), Smiley ([8]), Ru ([9]), Dethloft-Tan ([1]), Phuong ([6],[7]) and many auther. In this paper by using the second main theorem with ramification of Phuong-Thin (see [5]) we give some uniqueness results for linearly non-degenerate holomorphic curves on annulus sharing sufficiently many hyperplanes in projective space. First, we introduce some notations. Let R0 > 1 be a fixed positive real number or +∞, set 1 n, in Pn (C) are said to be in general position if for any distinct i1 , . . . , in+1 ∈ {1, . . . , .