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.