Lopez-Ros No-Go Theorem

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          About the Lopez-Ros No-Go Theorem

                            H. Karcher

   The theorem of  Lopez-Ros  [LR] says that a  complete,
minimal embedding of a punctured sphere is either a catenoid
or a plane.

    Our example is parametrized by a 3-punctured sphere, and
its Gauss map is Gauss(z) = cc(z-1)(z+1).  Parameter lines on
the sphere extend polar coordinates around the punctures at
z=+ee, z=-ee, z= ∞.

  A necessary condition for embeddedness  is parallel normals
at infinity, i.e.,   ee=1.  In this case the period cannot be closed.
If ee<>1, then cc can be chosen to close the period, but then
the catenoid  ends are tilted so that they intersect the third
(planar) end. -- For each ee<>1 we set cc0 to the value which
closes the period; one can therefore see in the morphing how
cc is used to close the period.

[LR]  F.J. Lopez and A. Ros,  On embedded complete minimal
       surfaces of genus zero, Journal of Differential Geometry
       33 (1), 1991, pp 293--300



  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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