8 HENRY C, WENTE
-0) -0)
k = 2He sinho), k = 2He cosho)
The k —curvature lines are parallel to the u-axis. Setting
(p(vi) = H would put the k -curvature lines parallel to the u-axis.
The Gauss equation (2.3) becomes
TT
2.
20)
-20),
Ao) + H (e - e ) = 0
(2.11)
2
Ao) + 4H sinho)cosho) = 0
From the form of (2.11) it is convenient to choose H = 1/2.
3
For the case of minimal immersions into R we select
0(w) = -1. This makes
-200 _ -20)
,
- ~ . . ~
K
e .
K
—•
e
(2.12)»
l 2
- (dx djf) = - du + dv
and the Gauss equation is now
(2.13) Aco - e~2a) = 0
which is the classical Liouville equation.
Finally, for cmc immersions with 0 H 1 into hyperbolic
3 / ST.
space M (-1) we select 0(w) = - yl - H This makes
k = H - yl - H e , k = H + yl - H e
(2.14) * 2
-dx-d? =
(He2a
- V l -
H2)du2
+
(He2(°
+ / l -H
2
)dv
2
and th e Gauss equatio n becomes
i r -, r- v k it
TT2
v ,
20)
—20).
(2.15) Ao)=(l-H)(e + e ).
For convenience we shall set H = 0.
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