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 tóm tắt luận án

f g C 5
f = g
1975,
C
n
P
N
(C)
f g C
n
P
N
(C) f g
(3N + 2)
P
N
(C), f ≡ g f g C
n
P
N
(C) (3N + 1)
P
N
(C) L P
N
(C)
g = L(f)
f C
n
P
N

(C)
H P
N
(C) ν
(f,H)
(z), z ∈ C
n
f H f(z)
z ∈ C
n
ν
(f,H),≤k
(z) =





0 ν
(f,H)
(z) > k,
ν
(f,H)
(z) ν
(f,H)
(z) ≤ k,
ν
(f,H),>k
(z) =






ν
(f,H)
(z) ν
(f,H)
(z) > k,
0 ν
(f,H)
(z) ≤ k.
k, d +∞ q H
1
, ··· , H
q
P
N
(C)
dim{z : ν
(f,H
i
),≤k
> 0 ν
(f,H
j
),≤k
> 0}  n − 2 1 ≤ i < j ≤ q
F


{H
j
}
q
j=1
, f, k, d)
g C
n
P
N
(C)
min{ν
(g,H
j
),≤k
(z), d} = min{ν
(f,H
j
),≤k
(z), d}, j ∈ {1, ··· , q}
k, d
g = f
q

j=1
{z : ν
(f,H
j
),≤k
(z) > 0}.

k = +∞ F

{H
j
}
q
j=1
, f, d)
C
n
P
N
(C)
q k, d F

{H
j
}
q
j=1
, f, k, d)
F

{H
j
}
q
j=1
, f, k, d)
P

N
(C)
q
d k
 F(f, {H
i
}
3N+2
i=1
, 1) = 1,
 F(f, {H
i
}
3N+1
i=1
, 1) = 1, N ≥ 2,
 F(f, {H
i
}
[2.75N]
i=1
, 1) = 1 N ≥ N
0
N
0
 F(f, {H
i
}
2N+3
i=1

, 1) = 1.
q < 2N + 3,
q ≤ 2N + 2
q = 2N + 2.
C
R
m
M R
3
,
R
3
.
G M p ∈ M
G(p) ∈ S
2
M p. G,
g := π ◦G : M → C := C ∪{∞}(= P
1
(C)) π S
2
P
1
(C).
z = u +
√
−1v
(u, v), M
ds
2

. M, g
M.
R
m
.
R
3
P
1
(C).
R
3
P
1
(C).
P
1
(C)
{z| 0 < 1/r < |z| < r}
R
3
P
1
(C)
m > 3.
R
m
(m ≥ 3)
m = 3; 4.
m > 3.

m = 4.
C
n
P
N
(C)
R
m
(m = 3; 4)
C
n
P
N
(C)
R
m
(m = 3; 4)
q = 2N + 2
R
m
R
3
, R
4
m = 3.
C
n
P
N

(C) P
N
(C)
q ≥ 3N + 2  F(f, {H
i
}
q
i=1
, 1) = 1.
N ≥ 2  F(f, {H
i
}
3N+1
i=1
, 1) = 1.
N
0
 F(f, {H
i
}
q
i=1
, 1) = 1 N ≥ N
0
q = [2.75N].
N ≥ 1  F(f, {H
i
}
2N+3
i=1

, 1) = 1.
g ∈ F(f, {H
i
}
2N+2
i=1
, N + 1)
α ∈ C (i, j) 1 ≤ i < j ≤ q,
(H
i
, f)
(H
j
, f)
= α
(H
i
, g)
(H
j
, g)
.
P
N
(C) 2N + 3.
2N + 3?
f
1
f
2

C
n
P
N
(C) (N ≥ 2) H
1
, , H
2N+2
P
N
(C)
dim{z ∈ C
n
: ν
(f
1
,H
i
)
(z) > 0 ν
(f
1
,H
j
)
(z) > 0} ≤ n − 2
1 ≤ i < j ≤ 2N + 2.
min{ν
(f
1

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

,H
j
),≥N
, 1} = min{ν
(f
2
,H
j
),≥N
, 1} (1 ≤ j ≤ 2N + 2),
f
1
≡ f
2
N ≥ 2,  F

{H
j
}
2N+2
j=1
, f, 1) ≤ 2.
q ≤ 2N + 2.
§1.2
( ) N = 1  F(f, {H
i
}
3N+1
i=1
, k, 2) ≤ 2 k ≥ 15.

( ) N ≥ 2  F(f, {H
i
}
3N+1
i=1
, k, 2) ≤ 2 k ≥ 3N + 3 +
4
N − 1
.
( ) N ≥ 4  F(f, {H
i
}
3N
i=1
, k, 2) ≤ 2 k > 3N + 7 +
24
N − 3
.
( ) N ≥ 6  F(f, {H
i
}
3N−1
i=1
, k, 2) ≤ 2 k > 3N + 11 +
60
N − 5
.
C
C
n

f : C
n
→ P
N
(C)
H P
N
(C)
N
(f,H)
(r) + m
f,H
(r) = T (r, f) (r > 1).
f : C
n
→ P
N
(C)
H
1
, , H
q
(q ≥ N + 1)
P
N
(C).
|| (q − N − 1)T (r, f) ≤
q

i=1

N
(N)
(f,H
i
)
(r) + o(T (r, f)).
f C
n
.








m

r,
D
α
(f)
f

= O(log
+
T (r, f)) (α ∈ Z
n
+

).
M
∗
n
C
n
M
∗
n
/C
∗
G A = (a
1
, a
2
, , a
q
)
q a
i
G. q ≥ r > s > 1. q A
(P
r,s
) r a
l(1)
, , a
l(r)
A
i
1

, , i
s
(1 ≤ i
1
< < i
s
≤ r) j
1
, , j
s
(1 ≤ j
1
< < j
s
≤ r)
{i
1
, , i
s
} = {j
1
, , j
s
} a
l(i
1
)
a
l(i
s

)
= a
l(j
1
)
a
l(j
s
)
.
G A =
(a
1
, , a
q
) q a
i
G A (P
r,s
) r, s
q ≥ r > s > 1 i
1
, , i
q−r+2
1 ≤ i
1
< < i
q−r+2
≤ q
a

i
1
= a
i
2
= = a
i
q−r+2
.
Φ
α
(F
0
, , F
M
) ≡ 0 |α| ≤
M(M − 1)
2
.
ν
([d])
:= min {ν
F
0
,≤k
0
, d} = min {ν
F
1
,≤k

1
, d} = ··· = min {ν
F
M
,≤k
M
, d}
d ≥ |α|, ν
Φ
α
(z
0
) ≥ min {ν
([d])
(z
0
), d − |α|}
z
0
∈ {z : ν
F
0
,≤k
0
(z) > 0} \ A, A ≥ 2
F
0
= ··· = F
M
≡ 0, ∞

H n − 1, ν
Φ
α
(z
0
) ≥ M, ∀ z
0
∈ H.
f : C
n
→ P
N
(C)
H
1
, H
2
, , H
q
q P
N
(C). k
j
≥ N − 1 (1 ≤
j ≤ q),










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)) .
Φ
α
= Φ
α
(F
j
0
0
c
, , F
j
0
M
c
) ≡ 0 c ∈ C, |α| ≤
M(M − 1)
2
, 2 ≥ |α|
0 ≤ i ≤ M,




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)).
2N +2
C
n
P
N
(C).
f
1
f
2
C
n
P
N
(C) (N ≥ 2) H
1
, , H
2N+2
P
N
(C)
dim{z ∈ C

n
: ν
(f
1
,H
i
)
(z) > 0 ν
(f
1
,H
j
)
(z) > 0} ≤ n − 2
1 ≤ i < j ≤ 2N + 2. m
m >

2N + 2
N + 1

2N + 2
N + 1

−2

.
min{ν
(f
1
,H

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

j
)
(z), ν
(f
2
,H
j
)
(z)} > N ν
(f
1
,H
j
)
(z) ≡ ν
(f
2
,H
j
)
(z) (mod m)
z ∈ (f
1
, H
j
)
−1
(0) (1 ≤ j ≤ 2N + 2).
f
1

≡ f
2
f
1
, f
2
, f
3
: C
n
−→ P
N
(C) {H
i
}
q
i=1
P
N
(C). d, k, k
1i
, k
2i
, k
3i
1 ≤ k
1i
, k
2i
, k

3i
≤ ∞ (1 ≤ i ≤ q) 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} d = 0
M = m d = min{k
ji
− m > 0 | 1 ≤ j ≤ 3; 1 ≤ i ≤ q} M = m.
dim{z ∈ C
n
: ν
(f
j
,H
i
),≤k
ji
> 0 ν
(f
j
,H
l
),≤k
jl
> 0} ≤ n − 2
(1 ≤ j ≤ 3; 1 ≤ i < l ≤ q)

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)
f
1
≡ f
j

q
α=1
{z ∈ C
n
: ν
(f
1
,H
α
),≤k

1α
(z) > 0} (1 ≤ j ≤ 3).
f
1
≡ f
2
f
2
≡ f
3
f
3
≡ f
1
N ≥ 2, 3N − 1 ≤ q ≤ 3N + 1, m > 3N + 1 +
16
3(N − 1)
(2q − 5N − 3) >
2Nk
m + 1
+
2N(q − k)
m + d + 1
−
3N
2
+ N
M + 1
.
N = 1, q = 4

3(2k + 1)
m + 1
+
6(4 − k)
m + d + 1
+
6k
M(m + 1)
+
24 − 6k
M(m + d + 1)
< 1 +
12
M
.
M = m k = q.
k = 1, M = m + d d = 1 d = 2,
f
1
, f
2
, f
3
: C
n
−→ P
N
(C) {H
i
}

3N+1
i=1
P
N
(C). k
i
1 ≤ i ≤ 3N + 1.
dim{z ∈ C
n
: ν
(f
j
,H
i
),≤k
i
> 0 ν
(f
j
,H
l
),≤k
l
> 0} ≤ n − 2(1 ≤ i < l ≤ 3N + 1)
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)
f
1
≡ f
j

3N+1
α=1
{z ∈ C
n
: ν
(f
1
,H
α
),≤k
α
(z) > 0} (1 ≤ j ≤ 3).
f
1
≡ f
2

f
2
≡ f
3
f
3
≡ f
1
N ≥ 2, k
j
= k
1
+ 1 2 ≤ j ≤ 3N + 1 k
1
> 3N + 2 +
14
3(N − 1)
.
N ≥ 2, k
j
= k
1
+ 2 2 ≤ j ≤ 3N + 1 k
1
> 3N + 1 +
16
3(N − 1)
.
k = 1 M = m +d
f

1
, f
2
, f
3
: C
n
−→ P
1
(C) {H
i
}
4
i=1
P
N
(C). k
i
(1 ≤ i ≤ 4)
dim{z ∈ C
n
: ν
(f
j
,H
i
),≤k
i
> 0 ν
(f

j
,H
l
),≤k
l
> 0} ≤ n − 2
1 ≤ j ≤ 3; 1 ≤ i < l ≤ 4
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
f
1
≡ f
j

4
α=1
{z ∈ C

n
: ν
(f
1
,H
α
),≤k
α
(z) > 0} (1 ≤ j ≤ 3)
k
1
= 9, k
2
= k
3
= k
4
= 66.
k
1
= 10, k
2
= k
3
= k
4
= 36.
k
1
= 11, k

2
= k
3
= k
4
= 26.
k
1
= 12, k
2
= k
3
= k
4
= 21.
k
1
= 13, k
2
= k
3
= k
4
= 18.
k
1
= 14, k
2
= k
3

= k
4
= 16.
f
1
≡ f
2
f
2
≡ f
3
f
3
≡ f
1
.
f C
n
P
N
(C) C
d k k = ∞. H
1
, , H
q
q
P
N
(C)
dim{z ∈ C

n
: ν
(f,H
i
)
(z) > 0 ν
(f,H
j
)
(z) > 0} ≤ n − 2 (1 ≤ i < j ≤ q),
G(f, {H
j
}
q
j=1
, k, d) g : C
n
→ P
N
(C)
g C,
min{ν
(f,H
j
),≤k
, d} = min{ν
(g,H
j
),≤k
, d} (1 ≤ j ≤ q),

f = (f
0
: ··· : f
N
) g = (g
0
: ··· : g
N
)
f g. 0  j  N ω ∈

q
i=1
{z ∈ C
n
: ν
(f,H
i
),k
(z) > 0},
f
j
(ω) = 0 g
j
(ω) = 0
f
j
(ω)g
j
(ω) = 0 D

α

f
i
f
j

(ω) = D
α

g
i
g
j

(ω) n α = (α
1
, , α
n
)
|α| = α
1
+ + α
n
 d i = j,
D
α
=
∂
|α|

∂
α
1
z
1
∂
α
n
z
n
.
N ≥ 4 2  d  N−1,  G(f, {H
i
}
3N+2−2d
i=1
, k, d) =
1 k >
3dN
2
− 2N
2
+ 2Nd − 2Nd
2
2(d − 1)N + d − 2d
2
− 1.
C
n
P

N
(C)
f, g : C
n
−→ P
N
(C) (N ≥ 2)
{a
j
}
3N+1
j=1
f
(f, a
i
) ≡ 0, (g, a
i
) ≡ 0 (1  i  3N + 1). f
R({a
j
}
3N+1
j=1
). d = 3N(N + 1)


2N+2
N+1



2


2N+2
N+1

− 1

+N(3N + 4).
dim{z ∈ C
n
: ν
(f,a
i
),d
(z) > 0 ν
(f,a
j
),d
(z) > 0}  n − 2
(1  i  N + 3, 1  j  3N + 1).
min{ν
(f,a
i
)
, d} = min{ν
(g,a
i
)
, d} ((1  i  3N + 1).

f(z) = g(z)

j∈D
{z ∈ C
n
: ν
(f,a
j
),M
(z) > 0}, D
{1, ··· , 3N + 1} D = N + 4
f ≡ g
D = N + 4 D < N + 4.
D N + 4 N + 2.
f, a : C
n
−→ P
N
(C) f = (f
0
:
··· : f
N
), a = (a
0
: ··· : a
N
) (f, a) =
N


i=0
a
i
f
i
, (f, a)(z) =
N

i=0
a
i
(z)f
i
(z).
a f T
a
(r) = o(T
f
(r)) r → ∞.
m
f,a
(r)
m
f,a
(r) =

S(r)
||f|| · ||a||
|(f, a)|
σ

n
−

S(1)
||f|| · ||a||
|(f, a)|
σ
n
,
a =

|a
0
|
2
+ ··· + |a
N
|
2

1/2
a
1
, . . . , a
q
(q ≥ N + 1) q C
n
P
N
(C)

a
j
= (a
j0
: ··· : a
jN
) (1  j  q).
a
1
, . . . , a
q
det(a
j
k
l
) ≡ 0 1  j
0
< j
1
< < j
N
 q.
M
n
C
n
R


a

j

q
j=1

⊂ M
n
C
a
jk
a
jl
a
jl
≡ 0.

R


a
j

q
j=1

⊂ M
n
h ∈ M
n
h

k
∈ R


a
j

q
j=1

k
f C
n
P
N
(C) f = (f
0
: ··· : f
N
)
R


a
j

q
j=1



R


a
j

q
j=1

f
0
, . . . , f
N
R


a
j

q
j=1


R


a
j

q

j=1

f a C
n
P
N
(C) z ∈ C
n
ν
(f,a),≤k
(z) =





0 ν
(f,a)
(z) > k,
ν
(f,a)
(z) ν
(f,a)
(z) ≤ k,
ν
(f,a),>k
(z) =






ν
(f,a)
(z) ν
(f,a)
(z) > k,
0 ν
(f,a)
(z) ≤ k.
f, a : C
n
→ P
N
(C)
(f, a) ≡ 0.
T (r, f) + T (r, a) = m
f,a
(r) + N
(f,a)
(r).
f : C
n
→ P
N
(C)
{a
j
}
q

j=1
(q ≥ N + 2) C
n
P
N
(C) f R({a
i
}
q
i=1
).
||
q
N + 2
T (r, f) ≤

q
j=1
N
(N)
(f,a
j
)
(r) + o(T (r, f)) + O(max
1≤j≤q
T (r, a
j
)).
k, d ∞


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

.
f, g : C
n
→ P
N
(C) (N ≥ 2) {a

j
}
3N+1
j=1
3N + 1 f C
n
P
N
(C)
dim{z ∈ C
n
: ν
(f,a
i
),k
(z)ν
(f,a
j
),k
(z) > 0}  n − 2 (1  i < j  3N + 1).
f, g R({a
j
}
3N+1
j=1
)
min (ν
(f,H
j
),k

, d) = min (ν
(g,H
j
),k
, d) (1  j  3N + 1).
f(z) = g(z)

j∈D
{z ∈ C
n
: ν
(f,a
j
),N(N+2)
(z) > 0}, D
{1, ··· , 3N + 1} D = N + 2
f ≡ g.
f, g : C
n
→ P
N
(C)
k k > 2N
3
+ 12N
2
+ 6N −1. {a
t
}
N+2

t=1
f C
n
P
N
(C)
dim{z ∈ C
n
: ν
(f,a
s
),k
(z)ν
(f,a
t
),k
(z) > 0}  n − 2 (1  s < t  N + 2).
f, g R({a
t
}
N+2
t=1
)
min (ν
(f,a
t
),k
, 1) = min (ν
(g,a
t

),k
, 1) (1  t  N + 2).
f = (f
0
: ··· : f
N
) g = (g
0
: ··· : g
N
) f
g 0  j  N ω ∈

N+2
t=1
{z ∈ C
n
:
ν
(f,a
t
),k
(z) > 0},
f
j
(ω) = 0 g
j
(ω) = 0
f
j

(ω)g
j
(ω) = 0 D
α

f
i
f
j

(ω) = D
α

g
i
g
j

(ω) n α = (α
1
, , α
n
)
|α| = α
1
+ + α
n
 2N i = j,
D
α

=
∂
|α|
∂
α
1
z
1
∂
α
n
z
n
.
f ≡ g.
M R
3
,
R
3
.
M G p ∈ M
M G(p) ∈ S
2
. π S
2
P
1
(C)
G g := π ◦ G : M → C := C ∪ {∞}(= P

1
(C)).
z = u +
√
−1v
(u, v), M ds
2
.
M g M.
M R
3
M,
{z| 0 < 1/r < |z| < r}.
R
m
(m > 3)
R
m
.
R
3
, R
4
R
m
(m > 4)
R
m
R
m

R
m
R
m
f ∆
R
:= {z ∈ C; |z| < R} P
1
(C),
0 < R < ∞. f = (f
0
: f
1
) ∆
R
||f|| := (|f
0
|
2
+ |f
1
|
2
)
1/2
, W (f
0
, f
1
) := f

0
f

1
− f
1
f

0
.
a
j
(1 ≤ j ≤ q) q P
1
(C).
a
j
= (a
j
0
: a
j
1
) |a
j
0
|
2
+ |a
j

1
|
2
= 1(1 ≤ j ≤ q).
F
j
:= a
j
0
f
1
− a
j
1
f
0
(1 ≤ j ≤ q).
f a = (a
0
:
a
1
) ∈ P
1
(C) e F := a
0
f
1
−a
1

f
0
e. f a, f
a ∞.
 > 0, C
1
µ
a
1
, ··· , a
q

∆ 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
)
q −2 −

q
j=1
1
m

j
> 0 f a
j
m
j
j(1 ≤ j ≤ q). C µ(> 1)
a
j
m
j
(1 ≤ j ≤ q)
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
)
∆
R
− A v := 0 ∆
R
∩ A A := {z ∈ ∆
R
; Π
q
j=1
F
j
= 0} v
∆
R
∆ log v ≥ v
2
v
∆

R
. v ∆ log v ≥ v
2
v(z) ≤
2R
R
2
− |z|
2
.
δ q −2 −

q
j=1
1
m
j
> qδ > 0 f
a
j
m
j
j(1 ≤ j ≤ q),
C
0
||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
.
f : C → P
1

(C)
a
1
, , a
q
∈ P
1
(C) f a
j
m
j
j (1 ≤ j ≤ q)
q

j=1
(1 −
1
m
j
) > 2.
f
g A P
m−1
(C)
H = {(w
0
: ··· : w
m−1
) ∈ P
m−1

(C) : a
0
w
0
+
+ a
m−1
w
m−1
= 0} e
(g, H) := a
0
g
0
+ + a
m−1
g
m−1
e, g
g = (g
0
: : g
m−1
). g H, g
H ∞.
M R
m
g M k− g(M)
k P
m−1

(C),
1 ≤ k ≤ m − 1. {H
j
}
q
j=1
P
m−1
(C).
g H
i
m
i
i
q

j=1
(1 −
k
m
j
) > (k + 1)(m −
k
2
− 1) + m
M g
R
3
.
q (q > 4) a

1
, , a
q
∈ P
1
(C) M
a
j
m
j
j,

q
j=1
(1 −
1
m
j
) ≤ 4.
g
g
R
3
M R
3
A
M, A {z| 0 < 1/r < |z| < r}
z q (q > 4) a
1
, , a

q
∈ P
1
(C)
M a
j
m
j
j A,

q
j=1
(1 −
1
m
j
) ≤ 4.
x = (x
1
, x
2
, x
3
, x
4
) : M → R
4
R
4
. Q

2
(C) := {(w
1
: : w
4
)|w
2
1
+ + w
2
4
= 0} ∈ P
3
(C).
M g : M → Q
2
(C). Q
2
(C)
P
1
(C) × P
1
(C). g
g = (g
1
, g
2
) M(g
l

: M → P
1
(C)). φ
i
:= ∂x
i
/dz i = 1, , 4,
g
1
g
2
g
1
=
φ
3
+
√
−1φ
4
φ
1
−
√
−1φ
2
, g
2
=
−φ

3
+
√
−1φ
4
φ
1
−
√
−1φ
2
.
M R
4
ds
2
= |φ|
2
(1 + |g
1
|
2
)(1 + |g
2
|
2
)|dz|
2
,
φ := φ

1
−
√
−1φ
2
M R
4
g = (g
1
, g
2
) M. A M, A
{z|0 < 1/r < |z| < r} z a
11
, , a
1q
1
, a
21
, , a
2q
2
q
1
+ q
2
(q
1
, q
2

> 2) P
1
(C).
g
l
≡ (l = 1, 2), g
l
a
lj
m
lj
j (l = 1, 2) A, γ
1
=

q
1
j=1
(1 −
1
m
1j
) ≤ 2, γ
2
=

q
2
j=1
(1 −

1
m
2j
) ≤ 2,
1
γ
1
− 2
+
1
γ
2
− 2
≥ 1.
g
1
g
2
g
2
≡ ,
g
1
a
1j
m
1j
j,
γ
1

=
q
1

j=1
(1 −
1
m
1j
) ≤ 3.
• C
n
P
N
(C)
2N + 2.
• C
n
P
N
(C)
•
R
m
(m = 3; 4)
C
n
P
N
(C)

{H
j
} 2N + 2.