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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