Planar Enneper

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                       About Planar Enneper

                             H. Karcher

   The surfaces Wavy Enneper,  Catenoid Enneper,  Planar Enneper,
and Double Enneper are finite total curvature minimal immersions
of the once or twice punctured sphere---shown with standard
polar coordinates. These surfaces illustrate how the different
types of ends can be combined in a simple way.

   The pure Enneper Surfaces (Gauss(z)=z^k ) and the Planar
Enneper Surfaces have been re-discovered many times, because
the members of the associate family are CONGRUENT surfaces
(as can be seen in the interesting associate family morphing!!) and
the Weierstrass integrals integrate to polynomial (respectively
rational) immersions.

  For this example ee is an integer valued parameter that
determines the degree of dihedral symmetry of the surface.
If you set ee greater than zero, then start with umax = 0.
Imagine this surface by starting with a plane minus a disk
and then observe how an Enneper end is attached.
 Formulas are taken 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 Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991

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