## Lopez-Ros No-Go Theorem

Additional images: Anaglyph, Anaglyph Wireframe, Parallel Stereo

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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