1. Introduction
Some theoretical and practical problems of research often boil down to the whole function and normality of meromorphic function. In 1907, P. Montel raised the concept of normal family. And nowadays, the theory of normal family has a good development. The core of the theory of normal family is the study of the format rule. It is possible to relate the normality of a family of the function and the function value for Nevanlinna value distribution theory in the 1930s. In 1975, Israel mathematician L.Z Alcman, starting from Marty formal rule, gave the new method of formal rule called Zalcman lemma. Then, in 1989, Pang Xuecheng improved Zalcman’s way, who got the Pang-Zalcman lemma, and he made it possible to study involving the formal rule of derived function.
In high-dimensional complex manifold holomorphic mappings about formal rule research, Tu Zhenhan firstly studied normality problems which involves the multiplicity of holomorphic mappings in 1999. Then, in 2014, Yang Liu, Caiyun Fang and Xuecheng Pang who use the high-dimensional Zalcman lemma of GAladro and S.G Krantz, generalize two family of meromorphic function normal rule, which was proved by Xiao-jun Liu, Lisan Hua and Pang Xuecheng, and they get the relative results about the holomorphic curves.
In value distribution and normal family theory of meromorphic functions, there is a well-known phenomenon named Bloch’s heuristic principle, that is to say, a family of meromorphic functions which have a property
in common
should be normal on D if this property
forces a meromorphic function on the whole complex plane
to be constant. Later, Zalcman (see [2] [3] ) formulated a more precise statement (known as Zalcman’s principle and Zalcman-Pang principle) to determine the normality for families of meromorphic functions.
In the case of holomorphic curves, there does still exist some similar phenomenon.
Firstly, Chen and Yan [4] proved the following result concerning uniqueness theorem for holomorphic curves.
Theorem CY. Let f and g be two nonconstant holomorphic curves from
to
, and
be q hyperplanes in
located in general position such that
and
“share”
with
,
. If
, then
.
Remark 1.1. Here, “share” means not only
, but also requires
on those points where
.
Later, in [5], Yang et al. considered the corresponding result in normal family theory of holomorphic curves and obtained.
Theorem YFP.
is a family of holomorphic curves from a domain
into
and
be
hyperplanes in
located in general position. Suppose that for each
,
,
,
. Then
is normal on D.
Obviously, the condition in Theorem CY is much stronger than those in Theorem YFP. Naturally, a question is posed that how to narrow the gap.
Liu et al. [6] used a special curve of f named derived curve to replace g and proved the following theorem.
Theorem LPY. Let
be a family of holomorphic maps of a domain
to
. Let
be hyperplanes in
located in general position, where
. Assume also that the following two conditions hold for every
:
1) If
, then
2) If
, then
, where
is constant,
.
Then
is normal on D.
In [1], Liu et al. remove the restriction of the first coefficients of hyperplanes of above theorem in the case
and prove the following theorem.
Theorem LP.
is a family of holomorphic maps of a domain
to
. Let
,
be hyperplanes in
located in general position, where
. Assume the following conditions hold for every
:
1)
if and only if
2) If
, then
, where
is a constant.
Then
is normal on D.
Unfortunately, the proof this proof of this theorem cannot be generalized to the
, and we get the main result of article is as follows:
Theorem 1.1. Let
be a family of holomorphic maps of a domain
to
. Let
and
be hyperplanes in
located in general position, where
Assume the following conditions hold for every
:
1)
if and only if
2) If
, then
, where
is a constant.
Then
is normal on D.
Before we continue to the proof of the main result, let us set some notation.
Throughout, D is a domain in
and
always represents the first coordinate hyperplane. We write
to indicate that the sequence
converges to f uniformly on compact subsets of D by the Fubini-Study metric on
. For a holomorphic curve
in D, the spherical
derivative of f at the point z is still denoted by
.
Frequently, given a sequence,
of maps, we need to extract an appropriate subsequence; and this necessity may recur within a single proof. To avoid the awkwardness of multiple indices, we again denote the extracted subsequence by
(rather than, say,
), and signal this operation by “taking a subsequence and renumbering, “or simply “renumbering”. The same convention applies to sequence of constants.
2. Definitions and Notations
At first, we recall some definitions and notations for
.
Let
be the N-dimensional complex projective space, and for any
,
,
if and only if there exists some
, such that
. The equivalence class of
is denoted by
and
.
Let
be hyperplanes in
which are given by
(1)
with nonzero coefficient vectors
Definition 2.1. The hyperplanes
are said to be in general position if for any injective maps
, the vectors
are linearly independent.
Second, let
be a holomorphic map and U be an open set in D. Any holomorphic map
such that
in U is called a representation of f on U, where
is the standard quotient map.
Definition 2.2. For any open subset U of D,
is called to be a reduced representation of f on U, if
are holomorphic functions on U without common zeros.
Let
be a hyperplane, we denote
. Throughout, we only consider normalized hyperplane representations so that
Next, for any reduced representation f of a holomorphic map f, we define the holomorphic function.
(2)
And put
(3)
Finally, according to the definition of the derived curves in [7], we have the following definition.
Definition 2.3. Let f be a holomorphic map of D into
,
be any reduced representation of f on D with
,
. Then
(4)
is called to be the µth derived holomorphic map of f, where
be a holomorphic function such that
and
,
have no common zeros.
Remark 2.1. For simplicity, we also write
as
, and obviously the definition of
does not depend on the choice of a reduced represention off.
3. Preliminary Lemmas
Before we give the proof of our main theorem, we need the following version of Zalcman’s lemma for holomorphic mappings from the domain
to
, which was proved by Aladro and Krantz in [8].
Lemma 3.1 (see [8] ). Let
be a family of holomorphic maps of a hyperbolic domain
into
. The family
is not normal on Ω if and only if there exist sequences
,
, with
,
and
,
, such that
converges uniformly on compact subsets of
to a nonconstant holomorphic map g of
into
.
The degenerate second main theorem in Nevanlina theory shows the following fact
Lemma 3.2 (see ( [9], p. 141)). Let
be a holomorphic map, and
be hyperplanes in
in general position. If for each
, either
is contained in
, or
omits
, then f is constant.
Lemma 3.3 (see ( [10], p. 34)). If
and
are meromorphic functions in the finite plane such that
Then we have
where
,
.
Lemma 3.4 (see ( [9], p. 124)). Let
be nowhere zero entire functions with
Consider the partition
such that i and j are in the same class
if and only if
for some nonzero constant
. Then
for any
.
Lemma 3.5 (see ( [1], p. 5)). Let
be a holomorphic curve with finite order and
, where
is an integer. And let
be hyperplanes in
located in general position, whose first coefficients
are nonzero, for all
. Let
be any reduced representation of g, if we denote
and assume that
and
, then we have g is linearly degenerate.
4. Proof of Theorem 1.1
Suppose
is not normal on D. Then by Lemma 3.1, there exist points
, positive numbers
and holomorphic maps
such that
where g is a nonconstant holomorphic map with finite order on
Let
be the reduced representation of g. Since
, are in generalposotion, without loss of generality, we may assume that the first coefficients of
are nonzero.
Case (A). g is linearly nondegenerate.
From the case (A) of theorem LP, we have
and
,
, then by lemma 3.5, g is linearly nondegenerate, a contradiction.
Case (B). g is linearly degenerate.
We can suppose
are linearly nondegenerate.
Since
or
, sofrom Lemma 3.5, we have
and g is a holomorphic map.
Then
is holomorphic in
. And we have
.
Then
. Specially, when
,
.
, without loss of generality,
.
Let
,
and
are constants.
Since
are linearly degenerate, then there exist a nonzero vector
such that
Let
Since
, then let
. It is a nonzero vector. Then
are linearly degenerate.
Claim. There exist an injective map
, such that
are linearly nondegenerate.
Proof of claim. Without loss of generality, we may assume that there exists some constants
, such that
Since
Then
Then we can let
(*)
Let
If
, which implies the equation set
Have untrivial solution, then there exist some constants
, which are not identical to zero, such that
i.e.
Then
are linear degenerate, a contradiction and
.
If
are linearly degenerate, then there exists some constants
, which are not identical to zero, such that
By (*), we have
which implies
are linearly degenerate , a contradiction. The claim is proven.
Then
are linearly nondegenerate, which implies
are different from easy other.
Since
be hyperplanes in
located in general position.
Then every five exist four differences in
If there exist
, such that
, without loss of generality, we can let
, so
Since
are linearly degenerate. Without loss of generality, let
But obviously, there are not exists four linearly nondegenerate in
, a contradiction. So
Since
,
are linearly degenerate, which mean that there exists two identical in
Then every five exists two identical in
.
If
are differences, then
equals one of
, let
. Similarly,
equals one of
, let
.
Then it is impossible to exist four differences in
, a contradiction, so
are linearly degenerate.
If
are linearly nondegenerate, then
Since every four linearly independent in
. Let
,
, then
Since
then exist
, such that
,
If
are different, so
equal one of
, let
,
,
,
, then it is obviously that there not exist three differences in
, a contradiction.
So
are linearly degenerate.
So there exist constants
which are not identical to zero, such that
1)
, then
and
can be linear representation by
.
So there exist constants
, such that
,
,
Then
Since
are linearly independent, then exist
, such that
If for such
,
, which implies that
, a contradiction. Then, we have
and
.
From lemma 3.2, without loss of generality, we may assume the first coefficient of
is nonzero, i.e.
. If the first coefficient of
are still nonzero, i.e.
, we have g omits
or
is contained in
.
Since
, we have all zeros of
are multiple. Thus, all zeros of
are multiple.
If for all
,
or
. By lemma 3.2, we have g is a constant curve, a contradiction.
Then, there exists some
, such that
and
. This implies that
a contradiction.
Thus, the first coefficient of
are still zero, i.e.
. Similarly, we have all zeros of
are multiple, we have all zeros of
are multiple.
If for all
,
or
,
. By lemma 3.2, we have g is a constant curve, a contradiction.
Then, there exist some
,
or 9, such that
and
. This implies that
a contradiction.
2)
, then
are not identical to zero.
a)
, a contradiction.
b)
, then
, which implies
, then
, a contradiction.
Thus,
is normal on D. The proof of Theorem 1.1 is finished
So in a word, we can make a conclusion as follows:
Theorem 1.1. Let
be a family of holomorphic maps of a domain
to
. Let
and
be hyperplanes in
located in general position, where
Assume the following conditions hold for every
:
1)
if and only if
2) If
, then
, where
is a constant.
Then
is normal on D.