About the Chen-Gackstatter Minimal Surface and its
Analogs with Higher Dihedral Symmetry
This surface is the first finite total curvature immersion
of a Riemann surface of genus >0 (here the square Torus).
It looks like an Enneper Surface with a handle added parallel to
its center saddle. This description determines the Gauss map
only up to a multiplicative constant (cc), which we took as the
morphing parameter. If this parameter is general then we get
a doubly periodic minimal immersion of the plane. The morphing
indicates how the period can be closed for one value of cc with the
intermediate value theorem. The resemblance with the standard
Enneper Surface is emphasized by using polar coordinates around
the puncture. The dd=3 surface is an analogue which can be viewed
as a higher order (120 degree symmetric) Enneper Surface with
a Y-shaped handle glued in. It was first published in a 1988
Vieweg Calendar by Polthier and Wohlgemuth.....
Formulas are from:
H. Karcher, Construction of minimal surfaces, in "Surveys in
Geometry", Univ. of Tokyo, 1989, and Lecture Notes No. 12,
SFB 256, Bonn, 1989, pp. 1--96.
For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].
[KWH] H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
the minimal surfaces that led to its discovery, in "Global Analysis
in Modern Mathematics, A Symposium in Honor of Richard Palais'
Sixtieth Birthday", K. Uhlenbeck Editor, Publish or Perish Press, 1993
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
Minimal xxx P I, Grundlehren der math. Wiss. v. 295