A uniqueness theorem for meromorphic mappings with hypersurfaces and without counting multiplicities

T¹p chÝ Khoa häc & C«ng nghÖ - Sè 1(45) Tập 1/N¨m 2008

74

A UNIQUENESS THEOREM FOR MEROMORPHIC MAPPINGS

WITH HYPERSURFACES AND WITHOUT COUNTING MULTIPLICITIES

Bùi Khánh Trình (Trường Đại học Xây dựng-Hà Nội)

1. Introduction

In 1926, R. Nevanlinna [6] showed that for two nonconstant meromorphic functions f

and g on the complex plane C , if they have the same inverse images for five distinct values

then f=g. In 1975, H. Fujimoto [4] generalized the above result to the case of meromorphic

mappings of m C into n CP . Since that time this problem has been studied intensively by H.

Fujimoto, W. Stoll, L. Smiley, G. Dethloff, D. D. Thai, T. V. Tan, S. Ji, S. D. Quang, M. Ru and

others. We would like to note that their results about uniqueness problem of meromorphic

mappings of m C into n CP have been still restricted to the case of hyperplanes. The aim of this

paper is to give a uniqueness theorem for the case of hypersurfaces.

2. Preliminaries

2.1. For m

z = (z1

...,zm

) ∈C , we set

1/ 2

2

1

m

j

j

z z

=

 

=    ∑ and define:

B r {z C z r} S r {z C z r

m m

( ) = ∈ : 〈 , ( ) = ∈ : =

( ) ( ) 1 1 2 2 1

( ), , log log

4

m m c c c c d dd z d z dd z υ σ

π

− − −

= ∂ −∂ = = ∧

Let F be a nonzero holomorphic function on m C . For a set 1

( ,..., ) α α α = n

of nonnegative

integers, we set 1 2 ... α α α α = + + + n

and

1

1

...

n

n

F

F

z z

α

α

α α

∂

=

∂ ∂

D . We define the map

( ) : max{ : ( ) 0 F

z m F z α

ν = = D for all α with α < ∈Ω m z } ( ) .

For each positive integer p (or +∞ ), we define the counting of F (multiplicities are truncated by p) by

[ ]

[ ]

2 1

1

( ) ( ): (1 )

r p

p F

F m

n t N r dt r

t

−

= < < +∞ ∫

where

[ ]

( )

( ) min{ , }.

F

p

F F

B t

n t p

ν

ν υ

∩

= ∫

for m≥ 2 and

[ ]( ) min{ ( ), } p

F

z t

n t z p ν

≤

=∑ for m = 1

2.2. Let f be a meromorphic map of m C into n CP . For arbitrary fixed homogeneous

coordinates 0

( :...: ) ω ωn

of n CP , we stake a reduced representation 0

( ,..., )

n

f f f =

ɶ

which means