About the Catenoid Fence
These singly periodic surfaces are parametrized (aa) by
rectangular tori; our lines extend polar coordinates around
the two punctures to the whole Torus. The surfaces look
like a fence of catenoids, joined by handles; they were made by
Karcher and Hoffman, responding to the suggestive skew 4-noids.
The morphing parameter aa is the modulus (a function of the
length ratio) of the rectangular Torus. Formulas are from [K2]
[K2] 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