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