ABSTRACT
This paper considers surfaces of constant mean curvature H
3 3
immersed in Euclidean space R (or more generally in M (c) the
simply connected spaces of constant curvature c) with the property
that one family of curvature lines are spherical. Such surfaces
are said to be of Enneper type. This geometrical problem is
reduced to finding solutions to the Gauss equation
Aoo +(H +c)e -Be = 0 where B is a positive constant
and such that co(u,v) satisfies the auxiliary condition
2x s ot(u)e + /9(u)e f o r s u i t a b l e f u n c t i o n s cx(u),/?(u) . The
u
2
nature of the solutions depend strongly on the sign of H + c.
Key words: constant mean curvature immersions, spherical lines of
curvature, elliptic Sinn-Gordon equation.
Note: Work on this paper was done in part while the author was a
guest of SFB 256 at University of Bonn, Germany and also with the
support of the National Science Foundation.
The author would like to express his appreciation to Ivan Sterling
for providing the computer-generated illustrations appearing at
the end of this paper and to Pamela Zawierucha for typing and
final preparation of this work.
Received by Editors December 23, 1990.
Previous Page Next Page