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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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