Luận văn: Giả khoảng cách tương đối Kobayashi

Z X
Z
x
0
∈ X V x
Z ϕ
1
, , ϕ
m
V
X ∩ V =

x ∈ V


ϕ
i
(x) = 0, i = 1, , m

.
X
a ∈ X X a
U Z U ∩X
X X
a ∈ X X
X X

X X
X
X = ∅
X x ∈ X
U x M π : M → U
U
π
S X U
π : M \ π
−1
(S) → U \ S
π : X

→ X
x ∈ X U x π
−1
(U)
U
α
X

π


U
α
: U
α
→ U .
X


X x ∈ X
π
−1
(x) x π
π : E → X
K r
p ∈ X E
p
:= π
−1
(p) K r E
p
p
p ∈ X U p
h : π
−1
(U) → U × K
r
h(E
p
) ⊂ {p}×K
r
,
h
p
h
p
: E
p

h
−→ {p} × K
r
proj
−−→ K
r
,
K (U, h)
K π : E → X E
X E
X (E, π, X)
E, X π
h
π : E → Y f : X → Y

f
−1
E, π

, X

f
−1
E :=

(x, e) ∈ X ×E; f(x) = π(e)

,
π


= pr
1

f
−1
E, π

, X

(E, π, X)
f
0 < r < ∞ D
r
=

z ∈ C, |z| < r

D
1
= D D
r
r D C
X x y X
(D, X) D X
p
0
= x, p
1
, , p
k

= y X
a
1
, a
2
, , a
k
D f
1
, , f
k
(D, X)
f
i
(0) = p
i−1
, f
i
(a
i
) = p
i
, ∀i = 1, , k.
α =

p
0
, , p
k
, a

1
, , a
k
, f
1
, , f
k

x y X
d
X
(x, y) = inf
α

k

i=1
ρ
D
(0, a
i
), α ∈ Ω
x,y

,
Ω
x,y
x y
X
d

X
: X ×X → R X
X
(α) =
k

i=1
ρ
D
(0, a
i
)
α
X d
X
(x, y) = ∞ x y
f : X → Y f
d
X
(x, y) ≥ d
Y
(f(x), f(y)) ∀x, y ∈ X.
d
X
X
f : D → X
X Y
d
X×Y


(x, y), (x

, y

)

=

d
X
(x, x

), d
Y
(y, y

)

x, x

∈ X y, y

∈ Y
D
n
= D × × D
d
D
n


(x
1
, , x
n
), (y
1
, , y
n
)

=
i

d
D
(x
i
, y
i
)

.
X
d
X
: X × X → R
X
d
X
X

d
X
(p, q) = 0 ⇐⇒ p = q, ∀p, q ∈ X.
X X d
X
X
X
ρ X d
X
ρ {x
n
} ⊂ X
ρ(x
n
, x) → 0 ⇐⇒ d
X
(x
n
, x) → 0 n → ∞.
d
X
ρ(x
n
, x) → 0 d
X
(x
n
, x) → 0 n → ∞
d
X

(x
n
, x) → 0 ρ(x
n
, x) → 0 n → ∞
s > 0 {x
n
} x
n
ρ x s
x
n
x γ
γ : [a, b] → X
t → ρ(γ(t), x) t
0
∈ [a, b]
ρ(γ(t
0
), x) = s y
n
= γ(t
0
) x
s ρ
d
X
(y
n
, x) ≤ d

X
(x
n
, x) → 0 n → ∞.
{y
n
} {y
n
k
} y
x s ρ
d
X
(y, x) = lim
n→∞
d
X
(y
n
k
, x) = 0,
y = x X

X F f (D
R
, X)
u =
f
∗
F

2
R
2
ds
2
R
D
R
u(0) > c > 0
g ∈ (D
R
, X)
g
∗
F
2
R
2
ds
2
R
c D
R
c
0
g = f ◦µ
r
◦ϕ ϕ D
R
µ

r
r 0 < r < 1 µ
r
(z) = rz, ∀z ∈ D
R
t ∈ [0, 1) f
t
∈ (D
R
, X)
f
t
(z) = f ◦µ
t
(z) = f(tz), ∀z ∈ D
R
.
u
t
=
f
∗
t
F
2
R
2
ds
2
R

=
(f ◦µ
t
)
∗
F
2
R
2
ds
2
R
.
u
t
(z) =
µ
∗
t
f
∗
F
2
µ
∗
t
(R
2
ds
2

R
)
(z)
µ
∗
t
ds
2
R
ds
2
R
(z) = µ
∗
t
(u(z))
(t
2
(R
2
− |z|
2
)
2
(R
2
− |tz|
2
)
2

.
U(t) = sup
z∈D
R
u
t
(z) = sup
z∈D
R
u(tz)
(t
2
(R
2
− |z|
2
)
2
(R
2
− |tz|
2
)
2
.
u
t
(z) t ∈ [0, 1) u
t
(z) → 0

D
R
sup
z∈D
R
u
t
(z) D
R
U(t) [0, 1) u
t
(0) = u(0).t
2
> ct
2
> 0
U(t) > c t
U(0) = 0 r ∈ (0, 1) U(r) = c
z
0
∈ D
R
c = sup
z∈D
R
u
t
(z)
D
R

D
R
ϕ D
R
0 z
0
g = f ◦µ
R
◦ϕ
ϕ
∗
(ds
2
R
) = ds
2
R
.
g
∗
F
2
R
2
ds
2
R
(z) =
(f ◦µ
R

◦ ϕ)
∗
F
2
R
2
ds
2
R
(z) =
ϕ
∗
(f ◦µ
R
)
∗
F
2
R
2
ds
2
R
(z)
= ϕ
∗
(u
r
(z)) = u
r

(ϕ(z)) ≤ c.
g
∗
F
2
R
2
ds
2
R
(0) = u
r
(ϕ(0)) = u
r
(z
0
) = c.

X E
X h : C → X
h
∗
E
2
≤ cdzdz
c > 0
X X
h : C → X
E X X
F

X
X
a F
X
≥ aE X
v
n
∈

T X
E(v
n
) = 1 F
X
(v
n
) <
1
n
F
X
D
R
n
R
n
lim
n→∞
R
n

= ∞ f
n
∈ (D
R
n
, X)
f

n
(0) = v
n
f

n
(0) df
n
(e) e =

∂
∂z

0
C 0 ||e||
n
e
ds
2
R
n
=

4R
2
n
dzdz
(R
2
n
− |z|
2
)
2
D
R
n
2
R
n
u
n
=
f
∗
E
2
R
2
n
ds
2
R

n
D
R
n
0
u
n
(0) =
E(f

n
(0))
2
R
2
n
.||e||
2
n
=
E(v
n
)
2
2
2
=
1
4
.

f
n
0 < c <
1
4
g
n
∈ (D
R
n
, X)
g
∗
n
E
2
≤ cR
2
n
ds
2
R
n
D
R
n
0
g
n
(D

R
n
) ⊂ f
n
(D
R
n
)
g
n
∈ (D
R
n
, X)
g
∗
n
E
2
≤ cR
2
n
ds
2
R
n
≤ cR
2
m
ds

2
R
m
, ∀n ≥ m,
F
m
=

g
n


D
R
m
, n ≤ m

m
F
1
=

g
n


D
R
1


X
h
1
∈ (D
R
1
, X).
F
2
=

g
n


D
R
2

h
2
∈ (D
R
2
, X).
h
k
∈ (D
R
k

, X), k = 1, 2, ,
h
k
h
k−1
h ∈ (C, X) h
k
g
∗
n
E
2
0

cR
2
n
ds
2
R
n

z=0
= 4cdzdz = 0
(h
∗
E
2
)
z=0

= lim
n→∞
(g
∗
E
2
)
z=0
= 4cdzdz = 0.
f
g
∗
n
E
2
≤ cR
2
n
ds
2
R
n
h
∗
E
2
≤ 4cdzd
z.
h h
∗

E
2
≤ dzdz
h : C → X 
X
Y
X Y
C(X, Y ) X Y
sup F ⊂ C(X, Y )
x
0
∈ X ε > 0 δ > 0 x ∈ X
d(x, x
0
) < δ
d(f(x), f(x
0
)) < ε f ∈ F.
F X F
x ∈ X
X Y
F
C(X, Y ) F C(X, Y )
F X
x ∈ X F
x
=

f(x)



f ∈ F

Y
X Y X
Y x, y ∈ X, x = y
U x V y Y
d
X

X ∩ U, X ∩ V

> 0.
X Y
X Y
X {x
n
} {y
n
} X
x
n
→ x ∈ ∂X y
n
→ y ∈ ∂X d
X
(x
n
, y
n

) → 0 x = y
{x
n
} {y
n
} X
x
n
→ x ∈ X, y
n
→ y ∈ X.
d
X
(x
n
, y
n
) → 0 n → ∞ x = y
H Y ϕ Y
f ∈ (D, X)
f
∗
(ϕH) ≤ H
D
,
H
D
D
H Y f (D, X)
f

∗
H ≤ H
D
.
⇒ x, y ∈ X x = y
d
X
(x, y) > 0.
X
x, y ∈ ∂X x = y
d
X
(x
n
, y
n
) → 0 n → ∞
⇒ x, y ∈ X
d
X
d
X
(x, y) = 0 X
x = y
x ∈ X y ∈ ∂X y ∈ X d
X
(x, s) y ∈
(x, s) y
n
→ y y

n
∈ (x, s) n d
X
(x
n
, x) → 0
x
n
∈ (x, s/2) d
X
(x
n
, y
n
) → 0
⇒ K Y
C > 0 f ∈ (D, X)
f
∗
(CH) ≤ H
D
f
−1
(K).
{f
n
} ⊂ (D, X) z
n
∈
f

−1
n
(K) ⊂ D |df
n
(z
n
)| → ∞ D
z
n
= 0
|df
n
(0)| → ∞ n → ∞.
K
f
n
(0) → y ∈ K.
U y Y U
D
m
r
k ∈ Z
+
z
k
∈ D n
k
∈ Z
|z
k

| <
1
k
f
n
k
(z
k
) ∈ U. (∗)
r < 1 f
n
(D
r
) ⊂ U
n ≥ n
0
(r) f
n
(0) → y {f
n


D
r
}
D
r
|df
n
(0)| → ∞ (∗)

y
k
= f
n
k
(0), x
k
= f
n
k
(z
k
).
z
k
x
k
U
x
k
→ x x = y
d
X
(x
k
, y
k
) ≤ d
D
(0, z

k
) → 0 k → ∞.
K
1
⊂ K
2
⊂ Y
∞

i=1
K
i
= Y K
i
= U
i
,
U
i
U
i
⊂ U
i+1
K
i
C
i
> 0
f
∗

(C
i
H) ≤ H
D
.
ϕ Y ϕ ≤ C
i
K
i
f
∗
(ϕH) ≤ H
D
H Y
⇒ ϕH
⇒ x, y ∈ X x = y
U =
H
(x, s), V =
H
(y, s)
s H
H x = y s > 0
H
(x, 2s) ∩
H
(y, 2s) = ∅ x

∈ U ∩ X y


∈ V ∩ Y
d
X
(x

, y

) ≥ d
H
(x

, y

) ≥ s > 0.
d
H
f ∈ (D, X)
d
X
≥ d
H
X Y

δ X
d
X
(p, q) ≥ δ(p, q), ∀p, q ∈ X
X Y
X
Y X Y (D, X)

(D, Y )
(D, X) (D, Y ) X
Y
Y n
f
n
: D → X z
n
∈ D


df
n
(z
n
)v


≥ n


v


v ∈ T
z
n
D. (∗)
D D z
n

= 0
X Y y ∈ Y f
n
(0) → y
(D, X) (D, Y )

f
n

f
0 f

n
(0) → f

(0) (∗) X
Y
X Y
X Y (D, X)
d
H
H Y

V R J : V → V R J
V
J
2
:= J ◦J = −Id.
J R V
V C


α + iβ

v := αv + βJ(v) = αv + βJv.
V C

v
1
, v
2
, , v
n

V R
V
R
J : V
R
→ V
R
v → Jv = iv
J V
R
V
V
R
R

v
1

, v
2
, , v
n
, Jv
1
, Jv
2
, , Jv
n

C
n
=

(z
1
, z
2
, , z
n
) : z
j
= x
j
+ iy
j
∈ C

= R

2n
=

(x
1
, y
1
, x
2
, y
2
, , x
n
, y
n
)

.
J : R
2n
→ R
2n
J

(x
1
, y
1
, x
2

, y
2
, , x
n
, y
n
)

=

− y
1
, x
1
, , −y
n
, x
n

.
J R
2n
M m M
0
2m
T
x
(M
0
) M

0
x T
x
(M)
M x
(U, h) M x
h : U → U

⊂ C
m
h = (h
1
, , h
n
)

h : U → R
2m

h(x) =

h
1
(x), h
1
(x), , h
m
(x), h
m
(x)


.
(U,

h) M
0
x

∂
∂z
1




x
, ,
∂
∂z
n




x

C T
x
(M).


∂
∂x
j




x
,
∂
∂y
j




x

n
j=1
R T
x
(M
0
).
J : T
x
(M
0
) → T

x
(M
0
)
v = α
1
.
∂
∂x
1




x
+ β
1
∂
∂y
1




x
+ + α
n
.
∂
∂x

n




x
+ β
n
∂
∂y
n




x
∈ T
x
(M
0
)
J
v
= (−β
1
)
∂
∂x
1





x
+ α
1
∂
∂y
1




x
+ + (−β
n
).
∂
∂x
n




x
+ α
n
∂
∂y
n





x
.
J T
x
(M
0
)
M 2n π : T M → M
J : T(M) → T(M) T (M)
M
∀ x ∈ M : J
x
= J


T
x
(M)
: T
x
(M) → T
x
(M)
R T
x
(M)

J M