About the Schoen No-Go Theorem
H. Karcher
R. Schoen's characterization of the catenoid says:
Any finite total curvature complete embedded minimal
surface which has TWO ends, is the catenoid.
Our example shows what happens if one tries---in spite
of Schoen's theorem---to add a handle to the catenoid. The
fundamental piece is similar to that of the catenoid fence,
except that the handle does not go outward to the neighbouring
catenoid but goes inward to meet its other half. However a gap
remains and as one tries to close it (by morphing with the
modulus, aa, of the underlying rectangular Torus) the surface
degenerates to look almost like two catenoids which move farther
apart as one tries to close the gap. We try to show this with the
suggested morphing; the deformation goes between rather extreme
surfaces where one has to adjust how far one computes into the end
and then also the size. The movie is still a bit jumpy.
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