tóm tắt luận án tiếng anh vấn đề duy nhất của ánh xạ phân hình vào không gian xạ ảnh và tính rẽ nhánh của ánh xạ gauss của mặt cực tiểu đầy

UNIVERSIT
´
E DE BRETAGNE OCCIDENTALE
And
HANOI NATIONAL UNIVERSITY OF EDUCATION
PHAM HOANG HA
On unicity problems of meromorphic mappings of C
n
into
P
N
(C) and the ramification of the Gauss maps of complete
minimal surfaces
Summery of Doctoral Thesis in Mathematics
Supervisors: Professor GERD DETHLOFF and Professor DO DUC THAI
Hanoi, May 3, 2013
i
Introduction
1. Motivation of the thesis
Unicity problems of meromorphic mappings under a conditions on the inverse im-
ages of divisors were studied firstly by R. Nevanlinna in 1925. He 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 generalized Nevanlinna’s results to the case of meromorphic
mappings of C
n
into P
N
(C). He showed that for two linearly nondegenerate meromor-
phic mappings f and g of C into P
N

(C), if they have the same inverse images counted
with multiplicities for 3N +2 hyperplanes in general position in P
N
(C) then f ≡ g and
there exists a projective linear transformation L of P
N
(C) onto itself such that g = L.f
if they have the same inverse images counted with multiplicities for 3N +1 hyperplanes
in general position in P
N
(C). After that, this problem has been studied intensively by
a number of mathematicans as H. Fujimoto, W. Stoll, L. Smiley, M. Ru, G. Dethloff -
T. V. Tan, D. D. Thai - S. D. Quang and so on.
Here we introduce the necessary notations to state the results.
Let f be a nonconstant meromorphic mapping of C
n
into P
N
(C) and H a hyperplane
in P
N
(C). Let k be a positive integer or k = ∞. Denote by ν
(f,H)
the map of C
n
into
Z whose value ν
(f,H)
(a) (a ∈ C
n

) is the intersection multiplicity of the image of f and
H at f(a).
For every z ∈ C
n
, we set
ν
(f,H),≤k
(z) =

0 if ν
(f,H)
(z) > k,
ν
(f,H)
(z) if ν
(f,H)
(z) ≤ k,
ν
(f,H),>k
(z) =

ν
(f,H)
(z) if ν
(f,H)
(z) > k,
0 if ν
(f,H)
(z) ≤ k.
1

Take a meromorphic mapping f of C
n
into P
N
(C) which is linearly nondegenerate
over C, a positive integer d, a positive integer k or k = ∞ and q hyperplanes H
1
, , H
q
in P
N
(C) located in general position with
dim{z ∈ C
n
: ν
(f,H
i
),k
(z) > 0 and ν
(f,H
j
),k
(z) > 0} ≤ n − 2 (1 ≤ i < j ≤ q),
and consider the set F(f, {H
j
}
q
j=1
, k, d) of all meromorphic maps g : C
n

→ P
N
(C)
satisfying the conditions
(a) g is linearly nondegenerate over C,
(b) min (ν
(f,H
j
),≤k
, d) = min (ν
(g,H
j
),≤k
, d) (1 ≤ j ≤ q),
(c) f(z) = g(z) on

q
j=1
{z ∈ C
n
: ν
(f,H
j
),≤k
(z) > 0}.
When k = ∞, for brevity denote F(f, {H
j
}
q
j=1

, ∞, d) by F(f, {H
j
}
q
j=1
, d). Denote
by  S the cardinality of the set S.
The unicity problem of meromorphic mappings means that one gives an estimate
for the cardinality of the set F(f, {H
j
}
q
j=1
, k, d). Some natural questions arise and we
state the followings
Question 1. The number of hyperplanes (fixed targets) in P
N
(C) which are used.
In particular, how about q?
Question 2. How about the truncated multiplicities (d and k) ?
Question 3. Whether the fixed targets (hyperplanes) can be generalized to moving
targets (moving hyperplanes) or hypersurfaces?
On the question 1 and 2, we list here some known results as
Smiley  F(f, {H
i
}
3N+2
i=1
, 1) = 1, Thai-Quang  F(f, {H
i

}
3N+1
i=1
, 1) = 1, N ≥ 2, Dethloff-
Tan  F(f, {H
i
}
[2.75N]
i=1
, 1) = 1 for N ≥ N
0
(where the number N
0
can be explicitly
calculated) and Chen-Yan  F(f, {H
i
}
2N+3
i=1
, 1) = 1.
When q < 2N + 3, there are some results which were given by Tan and Quang.
Those results lead us to the question.
What can we say about the unicity theorems with truncated multiplicities in the case
where q ≤ 2N + 2?
The first purpose of this thesis is to study these problems. Firstly, we will give a
new aspect for the unicity problem with q = 2N + 2, and we also study the unicity
theorems with ramification of truncations.
The second purpose of this thesis is to give some answers relative to the question
2
3. Our results are following the results of Ru, Dethloff-Tan, Thai-Quang.

On the other hand, there are many interesting unicity theorems for meromorphic
functions on C given by certain conditions of derivations. We would like to study the
unicity problems of such type in several complex variables for fixed and moving targets.
Parallel to the development of Nevanlinna theory, the value distribution theory of
the Gauss map of minimal surfaces immersed in R
m
was studied by many mathemat-
icans as R. Osserman, S. S. Chern, F. Xavier, H. Fujimoto, S. J. Kao, M. Ru and
others.
Let M now be a non-flat minimal surface in R
3
, or more precisely, a connected
oriented minimal surface in R
3
. By definition, the Gauss map G of M is the map
which maps each point p ∈ M to the unit normal vector G(p) ∈ S
2
of M at p.
Instead of G, we study the map g := π ◦ G : M → C := C ∪ {∞}(= P
1
(C)) for
the stereographic projection π of S
2
onto P
1
(C). By associating a holomorphic local
coordinate z = u +
√
−1v with each positive isothermal coordinate system (u, v), M
is considered as an open Riemann surface with a conformal metric ds

2
and by the
assumption of minimality of M, g is a meromorphic function on M. After that, we
can generalize to the definition of Gauss map of minimal surfaces in R
m
. So there are
many analogous results between the Gauss maps and meromorphic mappings of C into
P
N
(C). One of them is the small Picard theorem.
In 1965, R. Osserman showed that the complement of the image of the Gauss map
of a nonflat complete minimal surface immersed in R
3
is of logarithmic capacity zero in
P
1
(C). In 1981, a remarkable improvement was given by F. Xavier that the Gauss map
of a nonflat complete minimal surface immersed in R
3
can omit at most six points in
P
1
(C). In 1988, H. Fujimoto reduced the number six to four and this bound is sharp:
In fact, we can see that the Gauss map of Scherk’s surface omits four points in P
1
(C).
In 1991, S. J. Kao showed that the Gauss map of an end of a non-flat complete minimal
surface in R
3
that is conformally an annulus {z|0 < 1/r < |z| < r} must also assume

every value, with at most 4 exceptions. In 2007, Jin-Ru generalized Kao’s results for
the case m > 3.
On the other hand, in 1993, M. Ru studied the Gauss map of minimal surface in R
m
with ramification. That are generalizations of the above-mentioned results. A natural
question is that how about the Gauss map of minimal surfaces on annular ends with
ramification. The last purpose of this thesis is to answer to this question for the case
3
m = 3, 4. We refer to the work of Dethloff-Ha-Thoan for the case m > 4.
2. Aim of study
The aim of study is to study the unicity problems for meromorphic mappings of C
n
into P
N
(C) with fixed hyperplanes, moving hyperplanes and truncated multiplicities.
Besides, this thesis also studies the Gauss map of minimal surfaces in R
3
, R
4
on annular
ends with ramification.
3. Object and scope of study
As in motivation of the thesis above, the objects of the thesis are studying the
unicity problems for meromorphic mappings of C
m
into P
n
(C) and the ramification
of the Gauss map of minimal surfaces in R
3

, R
4
. In this thesis, the main purpose is
improving the recent known results.
4. Method of study
In order to solve the problems of the thesis, we use the study methods and tech-
niques of Complex Analysis, Nevanlinna theory, Riemann surfaces, Differential Geom-
etry and we introduce some new techniques.
5. The results and significance of the thesis
The thesis includes 3 chapters.
In chapter 1, we study the unicity theorems with truncated multiplicities of mero-
morphic mappings in several complex variables for few fixed targets. In particular,
we give a new unicity theorem for the above-mentioned first purpose of this thesis.
After that we study the unicity theorems with ramification of truncations which is an
improvement of Thai-Quang’s results. At the end of this chapter we give a unicity
theorem of meromorphic mappings with a conditions on derivations.
In chapter 2, we study the unicity theorems with truncated multiplicities of mero-
morphic mappings in several complex variables sharing few moving targets. In partic-
ular, we improve strongly the results of Dethloff- Tan before. Beside that, we also give
a unicity theorem of meromorphic mappings for moving targets with a conditions on
derivations.
In chapter 3, we recall the Gauss map of minimal surfaces in R
m
and we study the
ramification of the Gauss map on annular ends in minimal surfaces in R
3
, R
4
.
4

6. Structure of the thesis
The structure of this thesis includes an introduction, the references and 3 chapters
which are based on previous results. These three chapters are based on four articles
(two of them were published and the others are submitted).
Chapter 1: Unicity theorems with truncated multiplicities of meromorphic mappings
in several complex variables for few fixed targets.
Chapter 2: Unicity theorems with truncated multiplicities of meromorphic mappings
in several complex variables sharing small identical sets.
Chapter 3: Value distribution of the Gauss map of minimal surfaces on annular ends.
5
Chapter 1
Unicity theorems with truncated
multiplicities of meromorphic
mappings in several complex
variables for few fixed targets
The unicity theorems with truncated multiplicities of meromorphic mappings of C
n
into the complex projective space P
N
(C) sharing a finite set of fixed hyperplanes in
P
N
(C) has been studied intensively by H. Fujimoto, L. Smiley, S. Ji, M. Ru, D.D. Thai,
G. Dethloff, T.V. Tan, S.D. Quang, Z. Chen, Q. Yan and others. The unicity problem
has grown into a huge theory.
We report here briefly the unicity problems with multiplicities of meromorphic
mappings
Theorem A.(Smiley) If q ≥ 3N + 2 then  F(f, {H
i
}

q
i=1
, 1) = 1.
Theorem B.(Thai-Quang) If N ≥ 2 then  F(f, {H
i
}
3N+1
i=1
, 1) = 1.
Theorem C.(Dethloff-Tan)There exists a positive integer N
0
(which can be explic-
itly calculated) such that  F(f, {H
i
}
q
i=1
, 1) = 1 for N ≥ N
0
and q = [2.75N].
Theorem D.(Chen-Yan) If N ≥ 1 then  F(f, {H
i
}
2N+3
i=1
, 1) = 1.
Theorem E.(Tan) For each mapping g ∈ F(f, {H
i
}
2N+2

i=1
, N + 1), there exist a
constant α ∈ C and a pair (i, j) with 1 ≤ i < j ≤ q, such that
(H
i
, f)
(H
j
, f)
= α
(H
i
, g)
(H
j
, g)
.
6
Theorem F. (Quang) Let f
1
and f
2
be two linearly nondegenerate meromorphic
mappings of C
n
into P
N
(C) (N ≥ 2) and let H
1
, , H

2N+2
be hyperplanes in P
N
(C)
located in general position such that
dim{z ∈ C
n
: ν
(f
1
,H
i
)
(z) > 0 and ν
(f
1
,H
j
)
(z) > 0} ≤ n − 2
for every 1 ≤ i < j ≤ 2N + 2. Assume that the following conditions are satisfied.
(a) min{ν
(f
1
,H
j
),≤N
, 1} = min{ν
(f
2

,H
j
),≤N
, 1} (1 ≤ j ≤ 2N + 2),
(b) f
1
(z) = f
2
(z) on

2N+2
j=1
{z ∈ C
n
: ν
(f
1
,H
j
)
(z) > 0},
(c) min{ν
(f
1
,H
j
),≥N
, 1} = min{ν
(f
2

,H
j
),≥N
, 1} (1 ≤ j ≤ 2N + 2),
Then f
1
≡ f
2
.
Theorem G. (Quang) If N ≥ 2 then  F(f, {H
i
}
2N+2
i=1
, 1) ≤ 2.
In the first part of this chapter, we would like to study the unicity theorems for the
case q ≤ 2N + 2. In particular, we shall prove Theorem 1.2 (Ha-Quang) which gives a
new aspect of them in the first part of this chapter.
In 2006, Thai-Quang showed that
Theorem H. (Thai-Quang) (a) If N = 1, then  F(f, {H
i
}
3N+1
i=1
, k, 2) ≤ 2 for
k ≥ 15.
(b) If N ≥ 2, then  F(f, {H
i
}
3N+1

i=1
, k, 2) ≤ 2 for k ≥ 3N + 3 +
4
N − 1
.
(c) If N ≥ 4, then  F(f, {H
i
}
3N
i=1
, k, 2) ≤ 2 for k > 3N + 7 +
24
N − 3
.
(d) If N ≥ 6, then  F(f, {H
i
}
3N−1
i=1
, k, 2) ≤ 2 for k > 3N + 11 +
60
N − 5
.
The second part of this chapter studies the unicity problems of meromorphic map-
ping with ramification of truncations. We are going to improve Theorem G by Theorem
1.3 (Ha). In particular, we ramify truncations k
i
for each hyperplanes H
i
(1 ≤ i ≤ q),

and we then give its corollaries.
As far as we know, there are many interesting unicity theorems for meromorphic
functions on C given by the certain conditions of derivations. We will give a unicity
theorem of such type in several complex variables for fixed targets. That is a unicity
theorem with truncated multiplicities in the case where N + 4 ≤ q < 2N + 2. We will
prove Theorem 1.4 (Ha-Quang) in the last part of this chapter.
7
1.1 Basic notions and auxiliary results from Nevan-
linna theory
In this section, we recall some notions and auxiliary results from Nevanlinna theory. We
introduce the definition of the divisors on C
n
, the counting functions of the divisors, the
characteristic function, the proximity function. After that, we recall some results which
play essential roles in Nevanlinna theory as the first main theorem, the second main
theorem for hyperplanes, the logarithmic derivative lemma. We also introduce some
lemmas or propositions which are used for the proof of main results in this chapter.
1.1.19. Lemma. Suppose that Φ
α
(F
0
, , F
M
) ≡ 0 with |α| ≤
M(M − 1)
2
. If
ν
([d])
:= min {ν

F
0
,≤k
0
, d} = min {ν
F
1
,≤k
1
, d} = ··· = min {ν
F
M
,≤k
M
, d}
for some d ≥ |α|, then ν
Φ
α
(z
0
) ≥ min {ν
([d])
(z
0
), d−|α|} for every z
0
∈ {z : ν
F
0
,≤k

0
(z) >
0} \ A, where A is an analytic subset of codimension ≥ 2.
1.1.20. Lemma. Suppose that the assumptions in Lemma 1.1.19 are satisfied. If
F
0
= ··· = F
M
≡ 0, ∞ on an analytic subset H of pure dimension n−1, then ν
Φ
α
(z
0
) ≥
M, ∀ z
0
∈ H.
1.1.21. Lemma. Let f : C
n
→ P
N
(C) be a linearly nondegenerate meromorphic
mapping. Let H
1
, H
2
, , H
q
be q hyperplanes in P
N

(C) located in general position.
Assume that k
j
≥ N − 1 (1 ≤ j ≤ q). Then









q −N − 1 −
q

j=1
N
k
j
+ 1

T (r, f) ≤
q

j=1

1 −
N
k

j
+ 1

N
(N)
(f,H
j
),≤k
j
(r) + o(T (r, f)) .
1.1.22. Lemma. Assume that there exists Φ
α
= Φ
α
(F
j
0
0
c
, , F
j
0
M
c
) ≡ 0 for some
c ∈ C, |α| ≤
M(M − 1)
2
, 2 ≥ |α| and the assumptions in Lemma 1.1.19 are satisfied.
Then, for each 0 ≤ i ≤ M, the following holds:





N
(2−|α|)
(f
i
,H
j
0
),≤k
ij
0
(r)+M

j=j
0
N
(1)
(f
i
,H
j
),≤k
ij
(r) ≤ N(r, ν
Φ
α
) ≤ T (r)+

M

l=0
N
(
M(M −1)
2
)
(f
l
,H
j
0
),>k
lj
0
(r)+o(T (r)).
8
1.2 A unicity theorem with truncated multiplici-
ties of meromorphic mappings in several com-
plex variables sharing 2N + 2 hyperplanes
Theorem 1.2. (Ha-Quang) Let f
1
and f
2
be two linearly nondegenerate meromorphic
mappings of C
n
into P
N

(C) (N ≥ 2) and let H
1
, , H
2N+2
be hyperplanes in P
N
(C)
located in general position such that
dim{z ∈ C
n
: ν
(f
1
,H
i
)
(z) > 0 and ν
(f
1
,H
j
)
(z) > 0} ≤ n − 2
for every 1 ≤ i < j ≤ 2N + 2. Let m be a positive integer such that
m >

2N + 2
N + 1

2N + 2

N + 1

−2

.
Assume that the following conditions are satisfied.
(a) min{ν
(f
1
,H
j
)
, 1} = min{ν
(f
2
,H
j
)
, 1} (1 ≤ j ≤ 2N + 2),
(b) f
1
(z) = f
2
(z) on

2N+2
j=1
{z ∈ C
n
: ν

(f
1
,H
j
)
(z) > 0},
(c) min{ν
(f
1
,H
j
)
(z), ν
(f
2
,H
j
)
(z)} > N or ν
(f
1
,H
j
)
(z) ≡ ν
(f
2
,H
j
)

(z) (mod m) for all
z ∈ (f
1
, H
j
)
−1
(0) (1 ≤ j ≤ 2N + 2).
Then f
1
≡ f
2
.
1.3 A unicity theorem for meromorphic mapping
sharing few fixed targets with ramification of
truncations
Theorem 1.3. (Ha) Let f
1
, f
2
, f
3
: C
n
−→ P
N
(C) be three meromorphic mappings
and let {H
i
}

q
i=1
be hyperplanes in general position. Let d, k, k
1i
, k
2i
, k
3i
be the integers
with
1 ≤ k
1i
, k
2i
, k
3i
≤ ∞ (1 ≤ i ≤ q). We set M = max{k
ji
}, m = min{k
ji
} (1 ≤ j ≤
3, 1 ≤ i ≤ q), k = max{{i ∈ {1, 2 ··· , q} | k
ji
= m} | 1 ≤ j ≤ 3}. Define by d = 0 if
M = m and d = min{k
ji
− m > 0 | 1 ≤ j ≤ 3; 1 ≤ i ≤ q} if M = m. Assume that the
following conditions are satisfied
(i) dim{z ∈ C
n

: ν
(f
j
,H
i
),≤k
ji
> 0 and ν
(f
j
,H
l
),≤k
jl
> 0} ≤ n −2
(1 ≤ j ≤ 3; 1 ≤ i < l ≤ q)
(ii) min(ν
(f
j
,H
i
),≤k
ji
, 2) = min (ν
(f
t
,H
i
),≤k
ti

, 2) (1 ≤ j < t ≤ 3; 1 ≤ i ≤ q)
(iii) f
1
≡ f
j
on

q
α=1
{z ∈ C
n
: ν
(f
1
,H
α
),≤k
1α
(z) > 0} (1 ≤ j ≤ 3).
9
Then f
1
≡ f
2
or f
2
≡ f
3
or f
3

≡ f
1
if one of the following conditions is satisfied
1) N ≥ 2, 3N −1 ≤ q ≤ 3N + 1, m > 3N + 1 +
16
3(N − 1)
and
(2q −5N − 3) >
2Nk
m + 1
+
2N(q −k)
m + d + 1
−
3N
2
+ N
M + 1
.
2) N = 1, q = 4 and
3(2k + 1)
m + 1
+
6(4 − k)
m + d + 1
+
6k
M(m + 1)
+
24 − 6k

M(m + d + 1)
< 1 +
12
M
.
Before proving, we now give some corollaries that are given directly from Theorem 1.3.
*) Theorem G is deduced immediately from the theorem 1.3 by choosing M = m
and k = q .
*) When k = 1, M = m + d and d = 1 or d = 2 , by using the case 1 of Theorem
1.3, we have the following
Corollary 1. Let f
1
, f
2
, f
3
: C
n
−→ P
N
(C) be three meromorphic mappings and
let {H
i
}
3N+1
i=1
be hyperplanes in general position. Let k
i
be the positive integers with
1 ≤ i ≤ 3N + 1 satisfying the following conditions

(i) dim{z ∈ C
n
: ν
(f
j
,H
i
),≤k
i
> 0 and ν
(f
j
,H
l
),≤k
l
> 0} ≤ n −2 ( 1 ≤ i < l ≤ 3N + 1)
(ii) min(ν
(f
j
,H
i
),≤k
i
, 2) = min (ν
(f
t
,H
i
),≤k

i
, 2) (1 ≤ j < t ≤ 3; 1 ≤ i ≤ 3N + 1)
(iii) f
1
≡ f
j
on

3N+1
α=1
{z ∈ C
n
: ν
(f
1
,H
α
),≤k
α
(z) > 0} (1 ≤ j ≤ 3).
Then f
1
≡ f
2
or f
2
≡ f
3
or f
3

≡ f
1
if one of the following conditions is satisfied
a) N ≥ 2, k
j
= k
1
+ 1 for every 2 ≤ j ≤ 3N + 1 and k
1
> 3N + 2 +
14
3(N − 1)
.
b) N ≥ 2, k
j
= k
1
+ 2 for every 2 ≤ j ≤ 3N + 1 and k
1
> 3N + 1 +
16
3(N − 1)
.
*) When k = 1 and M = m + d, by using the proof for the case 2 of Theorem 1.3,
we have the following
Corollary 2. Let f
1
, f
2
, f

3
: C
n
−→ P
1
(C) be three meromorphic functions and let
{H
i
}
4
i=1
be hyperplanes in general position. Let k
i
(1 ≤ i ≤ 4) be the positive integers
satisfying the following conditions
(i) dim{z ∈ C
n
: ν
(f
j
,H
i
),≤k
i
> 0 and ν
(f
j
,H
l
),≤k

l
> 0} ≤ n −2
( 1 ≤ j ≤ 3; 1 ≤ i < l ≤ 4)
(ii) min(ν
(f
j
,H
i
),≤k
i
, 2) = min (ν
(f
t
,H
i
),≤k
i
, 2) (1 ≤ j < t ≤ 3; 1 ≤ i ≤ 4)
(iii) f
1
≡ f
j
on

4
α=1
{z ∈ C
n
: ν
(f

1
,H
α
),≤k
α
(z) > 0} (1 ≤ j ≤ 3)
Assume that one of the following conditions is satisfied
a) k
1
= 9, k
2
= k
3
= k
4
= 66.
b) k
1
= 10, k
2
= k
3
= k
4
= 36.
10
c) k
1
= 11, k
2

= k
3
= k
4
= 26.
d) k
1
= 12, k
2
= k
3
= k
4
= 21.
e) k
1
= 13, k
2
= k
3
= k
4
= 18.
f) k
1
= 14, k
2
= k
3
= k

4
= 16.
Then f
1
≡ f
2
or f
2
≡ f
3
or f
3
≡ f
1
.
1.4 A unicity theorem for meromorphic mapping
sharing few fixed targets with a conditions on
derivations
Take a meromorphic mapping f of C
n
into P
N
(C) which is linearly nondegenerate over
C, a positive integer d, a positive integer k or k = ∞ and q hyperplanes H
1
, , H
q
in
P
N

(C) located in general position with
dim{z ∈ C
n
: ν
(f,H
i
)
(z) > 0 and ν
(f,H
j
)
(z) > 0} ≤ n − 2 (1 ≤ i < j ≤ q),
and consider the set G(f, {H
j
}
q
j=1
, k, d) of all meromorphic maps g : C
n
→ P
N
(C)
satisfying the conditions
(a) g is linearly nondegenerate over C,
(b) min{ν
(f,H
j
),≤k
, d} = min{ν
(g,H

j
),≤k
, d} (1 ≤ j ≤ q),
(c) Let f = (f
0
: ··· : f
N
) and g = (g
0
: ··· : g
N
) be reduced representations of
f and g, respectively. Then, for each 0  j  N and for each ω ∈

q
i=1
{z ∈ C
n
:
ν
(f,H
i
),k
(z) > 0}, the following two conditions are satisfied:
(i) If f
j
(ω) = 0 then g
j
(ω) = 0,
(ii) If f

j
(ω)g
j
(ω) = 0 then D
α

f
i
f
j

(ω) = D
α

g
i
g
j

(ω) for each n-tuple α =
(α
1
, , α
n
) of nonnegative integers with |α| = α
1
+ + α
n
 d and for each
i = j, where D

α
=
∂
|α|
∂
α
1
z
1
∂
α
n
z
n
.
Remark that the condition (c) does not depend on the choice of reduced represen-
tations.
The last part of this chapter proves the following.
Theorem 1.4. (Ha-Quang) If N ≥ 4 and 2  d  N−1, then  G(f, {H
i
}
3N+2−2d
i=1
, k, d) =
1 for each k >
3dN
2
− 2N
2
+ 2Nd − 2Nd

2
2(d − 1)N + d −2d
2
− 1.
11
Chapter 2
Unicity theorems with truncated
multiplicities of meromorphic
mappings in several complex
variables sharing small identical
sets
The unicity theorems with truncated multiplicities of meromorphic mappings of C
n
into
the complex projective space P
N
(C) sharing a finite set of fixed (or moving) hyperplanes
in P
N
(C) have received much attention in the last few decades, and they are related
to many problems in Nevanlinna theory and hyperbolic complex analysis .
For moving targets and truncated multiplicites, the following results are best and
due to Dethloff-Tan. They proved the following.
Theorem of Dethloff-Tan Let f, g : C
n
−→ P
N
(C) (N ≥ 2) be two nonconstant
meromorphic mappings, and let {a
j

}
3N+1
j=1
be ”small” (with respect to f) meromorphic
mappings of C
n
into P
N
(C) in general position such that (f, a
i
) ≡ 0, (g, a
i
) ≡ 0 (1 
i  3N + 1) and f is linearly nondegenerate over R({a
j
}
3N+1
j=1
). Set M = 3N(N +
1)


2N+2
N+1


2


2N+2

N+1

−1

+N(3N +4). Assume that the following conditions are satisfied.
(i) dim{z ∈ C
n
: ν
(f,a
i
),M
(z) > 0 and ν
(f,a
j
),M
(z) > 0}  n − 2
(1  i  N + 3, 1  j  3N + 1).
(ii) min{ν
(f,a
i
)
, M} = min{ν
(g,a
i
)
, M} ((1  i  3N + 1).
(iii) f(z) = g(z) on

j∈D
{z ∈ C

n
: ν
(f,a
j
),M
(z) > 0}, where D is an arbitrary subset
of {1, ··· , 3N + 1} with D = N + 4.
12
Then f ≡ g.
We would like to emphasize here that the assumption D = N + 4 in the above-
mentioned theorem is essential in their proofs. It seems to us that some key techniques
in their proofs could not be used for D < N + 4.
The first main purpose of the present chapter is to give a unicity theorem with
truncated multiplicities of meromorphic mappings in several complex variables sharing
N + 2 moving targets. In particular, we prove Theorem 2.2 (Ha-Quang-Thai). It is an
improvement of the above-mentioned theorem of Dethloff-Tan.
In this chapter, we also would like to study the unicity problems of meromorphic
mappings in several complex variables for moving targets with conditions on deriva-
tions. We will prove Theorem 2.3 (Ha-Quang-Thai) in the last part of this chapter.
2.1 Preliminaries
Let a
1
, . . . , a
q
(q ≥ N + 1) be q meromorphic mappings of C
n
into P
N
(C) with reduced
representations a

j
= (a
j0
: ··· : a
jN
) (1  j  q). We say that a
1
, . . . , a
q
are located in
general position if det(a
j
k
l
) ≡ 0 for any 1  j
0
< j
1
< < j
N
 q. We also say that
a
1
, . . . , a
q
are located in pointwise general position if the hyperplanes a
1
(z), . . . , a
q
(z)

are in general position as a set of fixed hyperplanes at every point z ∈ C
n
.
Let M
n
be the field of all meromorphic functions on C
n
. Denote by R


a
j

q
j=1

⊂
M
n
the smallest subfield which contains C and all
a
jk
a
jl
with a
jl
≡ 0. Define

R



a
j

q
j=1

⊂
M
n
to be the smallest subfield which contains all h ∈ M
n
with h
k
∈ R


a
j

q
j=1

for
some positive integer k.
Let f be a meromorphic mapping of C
n
into P
N
(C) with reduced representation f =

(f
0
: ··· : f
N
). We say that f is linearly nondegenerate over R


a
j

q
j=1


R

a
j

q
j=1


if f
0
, . . . , f
N
are linearly independent over R



a
j

q
j=1

(

R


a
j

q
j=1

, respectively).
Let f, a be two meromorphic mappings of C
n
into P
N
(C) with reduced representa-
tions f = (f
0
: ··· : f
N
), a = (a
0
: ··· : a

N
) respectively. Put (f, a) =
N

i=0
a
i
f
i
. We say
that a is ”small” with respect to f if T (r, a) = o(T (r, f)) as r → ∞.
After that, we recall some results which play essential roles in Nevanlinna theory as
the first main theorem, the second main theorem for moving targets, the logarithmic
derivative lemma. We also introduce some lemmas or propositions which are used for
13
the proof of main results in this chapter.
2.2 A unicity theorem with truncated multiplicities
of meromorphic mappings in several complex
variables sharing few moving targets
In this section, we prove the following.
Theorem 2.2. (Ha-Quang-Thai) Let k be a positive integer or k = ∞ and d be a
positive integer or d = ∞ such that the following is satisfied

3
d + 1
+
6
k + 1

2N + 2

N + 1



2N + 2
N + 1

−2

<

N + 2
N(N + 2)(N(N + 2) + 1)
−
2N + 2
k + 1

.
Let f, g : C
n
→ P
N
(C) (N ≥ 2) be two nonconstant meromorphic mappings, and let
{a
j
}
3N+1
j=1
be ”small” (with respect to f) meromorphic mappings of C
n

into P
N
(C) in
general position such that dim{z ∈ C
n
: ν
(f,a
i
),k
(z)ν
(f,a
j
),k
(z) > 0}  n − 2 (1  i <
j  3N + 1).
Assume that f, g are linearly nondegenerate over R({a
j
}
3N+1
j=1
) and the following are
satisfied.
(i) min (ν
(f,H
j
),k
, d) = min (ν
(g,H
j
),k

, d) (1  j  3N + 1).
(ii) f(z) = g(z) on

j∈D
{z ∈ C
n
: ν
(f,a
j
),N(N+2)
(z) > 0}, where D be an arbitrary
subset of {1, ··· , 3N + 1} with D = N + 2.
Then f ≡ g.
2.3 A unicity theorem for meromorphic mapping
with a conditions on derivations
In the present section, we will prove the following.
Theorem 2.3. (Ha-Quang-Thai) Let f, g : C
n
→ P
N
(C) be two meromorphic map-
pings, and k be a positive integer with k > 2N
3
+12N
2
+6N −1. Let {a
t
}
N+2
t=1

be ”small”
(with respect to f) meromorphic mappings of C
n
into P
N
(C) in general position such
that
dim{z ∈ C
n
: ν
(f,a
s
),k
(z)ν
(f,a
t
),k
(z) > 0}  n − 2 (1  s < t  N + 2).
Assume that f, g are linearly nondegenerate over R({a
t
}
N+2
t=1
) and the following are
satisfied.
14
(i) min (ν
(f,a
t
),k

, 1) = min (ν
(g,a
t
),k
, 1) (1  t  N + 2).
(ii) Let f = (f
0
: ··· : f
N
) and g = (g
0
: ··· : g
N
) be reduced representations of f and g,
respectively. Then, for each 0  j  N and for each ω ∈

N+2
t=1
{z ∈ C
n
: ν
(f,a
t
),k
(z) >
0}, the following two conditions are satisfied:
(a) If f
j
(ω) = 0 then g
j

(ω) = 0,
(b) If f
j
(ω)g
j
(ω) = 0 then D
α

f
i
f
j

(ω) = D
α

g
i
g
j

(ω) for each n-tuple α =
(α
1
, , α
n
) of nonnegative integers with |α| = α
1
+ + α
n

 2N and for each
i = j, where D
α
=
∂
|α|
∂
α
1
z
1
∂
α
n
z
n
.
Then f ≡ g.
Remark that the condition (ii) in Theorem 2.3 does not depend on the choice of reduced
representations.
15
Chapter 3
Value distribution of the Gauss map
of minimal surfaces on annular ends
Let M be a non-flat minimal surface in R
3
, or more precisely, a connected oriented
minimal surface in R
3
. By definition, the Gauss map G of M is the map which maps

each point p ∈ M to the unit normal vector G(p) ∈ S
2
of M at p. Instead of G, we
study the map g := π ◦G : M → C := C ∪{∞}(= P
1
(C)) for the stereographic projec-
tion π of S
2
onto P
1
(C). By associating a holomorphic local coordinate z = u +
√
−1v
with each positive isothermal coordinate system (u, v), M is considered as an open
Riemann surface with a conformal metric ds
2
and by the assumption of minimality of
M, g is a meromorphic function on M.
In 1988, H. Fujimoto proved Nirenberg’s conjecture that if M is a complete non-flat
minimal surface in R
3
, then its Gauss map can omit at most 4 points, and the bound is
sharp. In 1991, S. J. Kao showed that the Gauss map of an end of a non-flat complete
minimal surface in R
3
that is conformally an annulus {z|0 < 1/r < |z| < r} must also
assume every value, with at most 4 exceptions.
On the other hand, in 1993, M. Ru studied the Gauss map of minimal surface in
R
m

with ramification. In this chapter, we shall study the Gauss map of minimal sur-
faces in R
3
, R
4
on annular ends with ramification. In particular, we prove Theorem
3.4.5, Theorem 3.4.6 (Dethloff-Ha). We would like to refer the case R
m
(m > 4) to
Dethloff-Ha-Thoan.
16
3.1 Minimal surface in R
m
We recall some basic facts in differential geometry. In particular, we recall the defini-
tions of minimal surfaces, curvature of a Riemann surface, the conditions for a surface
to be a minimal surface.
3.2 The Gauss map of minimal surfaces
In this section, we recall some notions on the Gauss map of minimal surfaces and the
relations of the Gauss map with the minimality properties of the surfaces.
3.3 Meromorphic functions with ramification
Let f be a nonconstant holomorphic map of a disc ∆
R
:= {z ∈ C; |z| < R} into P
1
(C),
where 0 < R < ∞. Take a reduced representation f = (f
0
: f
1
) on ∆

R
and define
||f|| := (|f
0
|
2
+ |f
1
|
2
)
1/2
, W (f
0
, f
1
) := f
0
f

1
− f
1
f

0
.
Let a
j
(1 ≤ j ≤ q) be q distinct points in P

1
(C). We may assume a
j
= (a
j
0
: a
j
1
) with
|a
j
0
|
2
+ |a
j
1
|
2
= 1(1 ≤ j ≤ q), and set
F
j
:= a
j
0
f
1
− a
j

1
f
0
(1 ≤ j ≤ q).
3.3.1 Definition. One says that the meromorphic function f is ramified over a point
a = (a
0
: a
1
) ∈ P
1
(C)with multiplicity at least e if all the zeros of the function F :=
a
0
f
1
− a
1
f
0
have orders at least e. If the image of f omits a, one will say that f is
ramified over a with multiplicity ∞.
3.3.2 Proposition. (Fujimoto) For each  > 0 there exist positive constants C
1
and µ
depending only on a
1
, ··· , a
q
and on  respectively such that

∆ log

||f||

Π
q
j=1
log(µ||f||
2
/|F
j
|
2
)

≥
C
1
||f||
2q−4
|W (f
0
, f
1
)|
2
Π
q
j=1
|F

j
|
2
log
2
(µ||f||
2
/|F
j
|
2
)
3.3.3 Lemma. Suppose that q − 2 −

q
j=1
1
m
j
> 0 and f is ramified over a
j
with
multiplicity at least m
j
for each j(1 ≤ j ≤ q). Then there exist positive constants C
and µ(> 1) depending only on a
j
and m
j
(1 ≤ j ≤ q) which satisfy that if we set

v :=
C||f||
q−2−

q
j=1
1
m
j
|W (f
0
, f
1
)|
Π
q
j=1
|F
j
|
1−
1
m
j
log(µ||f||
2
/|F
j
|
2

)
17
on ∆
R
− {F
1
F
q
= 0} and v = 0 on ∆
R
∩ {F
1
F
q
= 0}, then v is continuous on ∆
R
and satisfies the condition ∆ log v ≥ v
2
in the sense of distribution.
3.3.4 Lemma. (Generalied Schwarz’ Lemma) Let v be a nonnegative real-valued con-
tinuous subhamornic function on ∆
R
. If v satisfies the inequality ∆ log v ≥ v
2
in the
sense of distribution, then
v(z) ≤
2R
R
2

− |z|
2
.
3.3.5 Lemma. For every δ with q −2 −

q
j=1
1
m
j
> qδ > 0 and f is ramified over a
j
with multiplicity at least m
j
for each j(1 ≤ j ≤ q), there exists a positive constant C
0
such that
||f||
q−2−

q
j=1
1
m
j
−qδ
|W (f
0
, f
1

)|
Π
q
j=1
|F
j
|
1−
1
m
j
−δ
≤ C
0
2R
R
2
− |z|
2
.
3.3.6 Proposition. Let f : C → P
1
(C) be a holomorphic map. For arbitrary distinct
points a
1
, , a
q
∈ P
1
(C) and f is ramified over a

j
with multiplicity at least m
j
for each
j, (1 ≤ j ≤ q) satisfying
q

j=1
(1 −
1
m
j
) > 2.
Then f is constant.
3.4 The Gauss map of minimal surfaces with ram-
ification
3.4.1 Definition. One says that g of an open Riemann surface A into P
m−1
(C) is
ramified over a hyperplane H = {(w
0
: ··· : w
m−1
) ∈ P
m−1
(C) : a
0
w
0
+ +a

m−1
w
m−1
=
0}with multiplicity at least e if all the zeros of the function (g, H) := a
0
g
0
+ +a
m−1
g
m−1
have orders at least e, where g = (g
0
: : g
m−1
). If the image of g omits H, one will
say that g is ramified over H with multiplicity ∞.
3.4.2 Theorem. (Ru) For any complete minimal surface M immersed in R
m
and
assume that the Gauss map g of M is k−nondegenerate (that is g(M) is contained in
a k−dimensional linear subspace of P
m−1
(C), but none of lower dimension ), 1 ≤ k ≤
m −1. Let {H
j
}
q
j=1

be hyperplanes in general position in P
m−1
(C). If g is ramified over
H
i
with multiplicity at least m
i
for each i and
q

j=1
(1 −
k
m
j
) > (k + 1)(m −
k
2
− 1) + m
18
then M is flat, or equivalently, g is constant.
3.4.3 Theorem. (Ru) Let M be a non-flat complete minimal surface in R
3
. If there
are q (q > 4) distinct points a
1
, , a
q
∈ P
1

(C) such that the Gauss map of M is ramified
over a
j
with multiplicity at least m
j
for each j, then

q
j=1
(1 −
1
m
j
) ≤ 4.
3.4.4 Corollary. The Gauss map g assumes every value on unit sphere with the possible
exception of four values.
3.4.5 Theorem. (Dethloff-Ha) Let M be a non-flat complete minimal surface in R
3
and let A be an annular end of M which is conformal to {z| 0 < 1/r < |z| < r}, where
z is the conformal coordinate. If there are q (q > 4) distinct points a
1
, , a
q
∈ P
1
(C)
such that the Gauss map of M is ramified over a
j
with multiplicity at least m
j

for each
j on A, then

q
j=1
(1 −
1
m
j
) ≤ 4.
3.4.6 Theorem. (Dethloff-Ha) Suppose that M is a complete non-flat minimal surface
in R
4
and g = (g
1
, g
2
) is the Gauss map of M. Let A be an annular end of M which
is conformal to {z|0 < 1/r < |z| < r}, where z is the conformal coordinate. Let
a
11
, , a
1q
1
, a
21
, , a
2q
2
be q

1
+ q
2
(q
1
, q
2
> 2) distinct points in P
1
(C).
(i) In the case g
l
≡ constant (l = 1, 2), if g
l
is ramified over a
lj
with multiplicity at
least m
lj
for each j (l = 1, 2) on A, then
γ
1
=

q
1
j=1
(1 −
1
m

1j
) ≤ 2, or γ
2
=

q
2
j=1
(1 −
1
m
2j
) ≤ 2, or
1
γ
1
− 2
+
1
γ
2
− 2
≥ 1.
(ii) In the case where one of g
1
and g
2
is constant, say g
2
≡ constant, if g

1
is
ramified over a
1j
with multiplicity at least m
1j
for each j, we have the following
γ
1
=
q
1

j=1
(1 −
1
m
1j
) ≤ 3.
3.4.7 Corollary.(Kao) The Gauss map g of minimal surfaces in R
3
on an annular
end must assume every value on unit sphere with the possible exception of four values.
19
CONCLUSION AND RECOMMENDATIONS
Conclusions
The main results of the thesis are the followings
• Unicity theorems with truncated multiplicities of meromorphic mappings of C
n
into P

N
(C) for fixed hyperplanes, truncated multiplicities and a small set of
identity or with the conditions of derivations.
• Unicity theorems with truncated multiplicities of meromorphic mappings of C
n
into P
N
(C) for moving targets, and a small set of identity or with the conditions
of derivations.
• The ramification of the Gauss map of minimal surfaces in R
m
(m = 3; 4) on an
annular end.
Recommendations on further research
1. What can we say about the unicity theorems of meromorphic mappings of C
n
into P
N
(C) for q(< 2N + 2) hyperplanes?
2. Can we give some unicity theorems of meromorphic mappings of C
n
into P
N
(C)
for moving targets with the conditions as in the case for hyperplanes.
3. Can we obtain on unicity theorems for the Gauss maps of minimal surfaces in
R
m
?
However, due to limited time we could not get the results to these problems. We

hope that these problems will soon be resolved in the near future.
20