Lopez-Ros No-Go Theorem

minimal surface
          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

See also
References on Mathematical Surface