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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