ENNEPER TYPE IMMERSIONS 7 (2.5) k = H - |0|e~2C° , k = H + |0| e~20) assuming that k k . the extrinsic Gauss curvature is defined 1 2 to be K = k k and the intrinsic Gauss curvature e 12 (2.6) K = K + c = ( H 2 + c ) - |0|2e~"4a) e . " ' Note: If a new set of coordinates is introduced by an analytic mapping w = f(w' ) then one obtains a new pair {$,0)}. They are related to the original pair by (2.7) . 2(0 200,_, , ,2 a) e = e f (W ) b) $(w') = 0(w) f (w' )2 From the latter equation we see that (2.8) $(w' )dw'2 = 0(w)dw2 This is the Hopf quadradric diferential [9]. From (2.4) we see that the zeros of /{w) correspond to the umbilic points of the immersion. If £(w) = 0 then all points are umbilic and the immersed surface is a sphere. Otherwise p(w) % 0 and the umbilic points are isolated. For a simply connected region where the Hopf differential does not vanish one can make a conformal change of coordinates so that £(w) = constant £ 0. The lines of curvature are given by (2.9) lm[0(w)dw2] = - Mdu2 + (L -N)dudv + Mdv2 = 0 from which it follows that the preimage of the lines of curvature are straight lines in the parameter domain. By a rotation one can make M = 0, p(vf) a real constant, and the lines of curvature parallel to the coordinate axes. 3 For a cmc immersion into R with H positive we may set 0(w) = -H. This implies
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