1-9 Equivalence theorem of conformal mapping 49
Theorem 1.12 (Conformeomorphism M
2
l → M
2
r , conformal mapping).
An orientation preserving conformal mapping M
2
l → M
2
r can be constructed by three steps in solving
special Korn–Lichtenstein equations.
1st step or left step.
The left Riemann manifold M
2
l
U
1 , U
2 G l
, which is called left surface, is parameterized by general
left parameters (general left coordinates) {U
1 , U
2
} or {U, V }. The solution of the following special
Korn–Lichtenstein equations (i), subject to the following integrability conditions of harmonicity (ii)
and orientation conservation (iii), is needed.
(i) Special KL:
P U
P V
=
1
√
G 11 G 22 −G 2
12
−G 12 G 11
−G 22 G 12
Q U
Q V
,
⎡
⎢
⎣
P U =
1
√
G 11 G 22 −G 2
12
(−G 12 Q U + G 11 Q V )
P V =
1
√
G 11 G 22 −G 2
12
(−G 22 Q U + G 12 Q V )
⎤
⎥
⎦ . (1.169)
(ii) Left integrability:
P UV = P V U and Q UV = Q V U
(1.170)
or (in terms of the Laplace–Beltrami operator)
⎡
⎢
⎢
⎢
⎢
⎣
∆ UV P :=
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
+
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
= 0
∆ UV Q :=
G 11 Q V −G 12 Q U
√
G 11 G 22 −G 2
12
V
+
G 22 Q U −G 12 Q V
√
G 11 G 22 −G 2
12
U
= 0
⎤
⎥
⎥
⎥
⎥
⎦
.
(1.171)
(iii) Left orientation conservation:
P U P V
Q U Q V
= P U Q V − P V Q U > 0 .
(1.172)
Note that the coordinates P and Q are the left conformal coordinates, which are also called isometric
or isothermal.
2nd step or right step.
The right Riemann manifold M
2
r
u
1 , u
2 G r
, which is called right surface, is parameterized by general
right parameters (general right coordinates) {u
1 , u
2
} or {u, v}. The solution of the following special
Korn–Lichtenstein equations (i), subject to the following integrability conditions of harmonicity (ii)
and orientation conservation (iii), is needed.
(i) Special KL:
p u
p v
=
1
√
g 11 g 22 −g 2
12
−g 12 g 11
−g 22 g 12
q u
q v
,
⎡
⎢
⎣
p u =
1
√
g 11 g 22 −g 2
12
(−g 12 q u + g 11 q v )
p v =
1
√
g 11 g 22 −g 2
12
(−g 22 q u + g 12 q v )
⎤
⎥
⎦ .
(1.173)