Chen Gackstatter

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         About the Chen-Gackstatter Minimal Surface and its
                Analogs with Higher Dihedral Symmetry

                         H. Karcher

     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
           Springer-Verlag, 1991

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