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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 P

n(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 mul￾tiplicity 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: Unicity, annuli, hyperplane, holomorphic curve, general position.

1 INTRODUCTION

In 1926, R. Nevanlinna proved that two non￾constant meromorphic functions of one com￾plex variable which attain same five distinct

values at the same points, must be identical. In

1975, H.Fujimoto (see [2]) generalized Nevan￾linna’s result to the case of meromorphic map￾pings of C

m into P

n(C). He given the suf￾ficient condition with 3n + 2 hyperplanes in

general position which determining a meromor￾phic maps. Since that time, this problem has

been studied intensively. The many mathe￾maticians study two following problems: find￾ing properties of unique range sets, and find￾ing out a unique range set with the smallest

number of elements as possible. For exam￾ple: 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 pro￾jective space. First, we introduce some nota￾tions.

Let R0 > 1 be a fixed positive real number or

+∞, set

∆ = 

z ∈ C :

1

R0

< |z| < R0

,

be an annulus in C. Let f : ∆ → P

n(C) be a

holomorphic curve, and f = (f0, . . . , fn) be a

reduced representative of f, namely f0, . . . , fn

are holormorphic functions on ∆ without com￾mon zeros. For 1 < r < R0, characteristic func￾tion Tf (r) of f is defined by

Tf (r) = 1

2π

Z 2π

0

log kf(reiθ)kdθ

+

1

2π

Z 2π

0

log kf(r

−1

e

iθ)kdθ,

where kf(z)k = max{|f0(z)|, . . . , |fn(z)|}. The

above definition is independent, up to an ad￾ditive constant, of the choice of the reduced

representation of f.

Let f = (f0, . . . , fn) : ∆ → P

n(C) be a holo￾morphic map where f0, . . . , fn are holomorphic

functions and without common zeros in ∆, let

H be a hyperplane in P

n(C) and

L(z0, . . . , zn) = Xn

j=0

aizj

be the linear form in n + 1 variables with co￾efficients in C defining H, where aj ∈ C, j =

0, . . . , n, be constants. Set

(f, H) = Xn

j=0

ajfj .

0*Tel: 0977615535, e-mail: nvphuongt@gmail.com