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}
or
{u, v}
, which parameterize the right surface, are
transformed to alternative right conformal coordinates
{p, q}
, which are also called
isometric
or
isothermal
. Indeed, the right differential invariant
I
r
∼
d
s
2
=
λ
2
(d
p
2
+d
q
2
) is described by
identical metric coecients
g
pp
=
g
qq
=
λ
2
and
g
pq
= 0. Third, the left conformal coordinates
{
Second, the original coordinates
{u
1
,u
2
are transformed to right conformal coordinates by solving the special Korn-Lichtenstein
equations for
P,Q
}
l
G
l
=
Λ
2
I
2
}→
M
r
G
r
=
λ
2
I
2
}
. which are called
Cauchy-Riemann
(
d
'
Alembert-Euler
)
equations
, subject to an integrability condition. The integrability condition
turns out to be the
vector-valued Laplace equation of harmonicity
, as stated in the following
theorem and proven later on.
M
{
P,Q
|
{
p, q
|
2
l
2
Theorem 1.12 (Conformeomorphism
M
→
M
r
, conformal mapping).
l
r
can be constructed by three steps in
An orientation preserving conformal mapping
M
→
M
solving special Korn-Lichtenstein equations.
1ststeporleftstep.
2
l
(
U
1
,U
2
The left Riemann manifold
G
l
), which is called
left surface
, is parameterized by
general
left parameters
(general left coordinates)
M
|
U
1
,U
2
. 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.
{
}
or
{
U, V
}
(i) Special KL:
P
U
P
V
=
−
Q
U
Q
V
,
1
G
11
G
22
−
G
12
G
11
−
G
22
G
12
G
12
⎡
⎤
√
G
11
G
22
−G
12
1
P
U
=
(
−
G
12
Q
U
+
G
11
Q
V
)
⎣
⎦
.
(1.169)
√
G
11
G
22
−G
12
1
P
V
=
(
−
G
22
Q
U
+
G
12
Q
V
)
(ii) Left integrability:
P
UV
=
P
VU
and
Q
UV
=
Q
VU
(1.170)
or (in terms of the Laplace-Beltrami operator)
⎡
Δ
UV
P
:=
G
11
P
V
−G
12
P
U
+
G
22
P
U
−G
12
P
V
⎤
√
G
11
G
22
−G
12
√
G
11
G
22
−G
12
=0
⎣
⎦
Δ
UV
Q
:=
G
11
Q
V
−
G
12
Q
U
+
G
22
Q
U
−G
12
Q
V
V
U
.
(1.171)
√
G
11
G
22
−G
12
√
G
11
G
22
−G
12
=0
V
U
(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
.
2ndsteporrightstep.
The right Riemann manifold
M
|
G
r
), which is called
right surface
, is parameterized by
general right parameters
(general right coordinates)
r
(
u
1
,u
2
u
1
,u
2
. The solution of the follow-
ing special Korn-Lichtenstein equations (i), subject to the following integrability conditions of
harmonicity (ii) and orientation conservation (iii), is needed.
{
}
or
{
u,v
}
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