Differential Equations, Mechanics, and Computation

H. Karcher

These examples generalize Scherk's conjugate pair of
singly-periodic/doubly-periodic minimal surfaces. The
singly-periodic examples stay embedded if the dihedral
symmetry (and with it the number of punctures) is increased
(Gauss(z)=z^k , k=ee-1). The most symmetric ones (bb=0.5/ee)
can be deformed by decreasing bb.
See [K1],  [K2] for more details.

These surfaces are parametrized by punctured spheres, but
the Weierstrass integrals have periods, a vertical one in the
singly periodic case, two horizontal ones for doubly periodic
surfaces. The parameter lines extend polar coordinates around
the punctures to the whole sphere---in these cases giving level
lines on the surfaces.

The degree of dihedral symmetry is, of course, a discrete
property,  and it is controlled by the parameter ee. Thus, ee
should be set to an integer  (the default is 2). For each choice
of ee,  changing bb gives a one-parameter family of surfaces,
of which the most symmetric member is obtained by setting
bb = 0.5/ee.  Try setting ee to 3 and 4, and bb to 0.333 and
0.25 respectively.  The wings of the singly periodic SaddleTower
surfaces become *parallel in pairs* if (for ee>2 ) one sets
bb = 0.0825.  These stay embedded for ee=3 and ee=4.

We also recommend viewing the associate family morphing.

[K1]  H. Karcher, Embedded minimal surfaces derived from Scherk's
examples, Manuscripta Math. 62 (1988) pp. 83--114.

[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
Springer-Verlag, 1991

We also recommend viewing the associate family morphing.

[K1]  H. Karcher, Embedded minimal surfaces derived from Scherk's
examples, Manuscripta Math. 62 (1988) pp. 83--114.

[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
Springer-Verlag, 1991