Karcher JE Saddle Tower

JE morph thin 001
These doubly periodic embedded minimal surfaces are parametrized by rectangular tori. The shape can be viewed as a doubly periodic version of Riemann's minimal surfaces.
je conj 2per morph 001
je conj 2per morph 001
je conj thin 001
je conj thin 001
karcher je saddle 2per pat
karcher je saddle 2per pat
karcher je saddle 2per wir
karcher je saddle 2per wir
JE wire anaglyph
JE wire anaglyph
JE conjugate anaglyph
JE conjugate (anaglyph) These conjugate surfaces are also 2-periodic and embedded. The shape can be viewed as an array of Scherk saddle towers, seperated by straight lines (symmetry lines of these surfaces). This picture led to their discovery.
       About the Doubly  Periodic JE Minimal Surfaces

                           H. Karcher

   These families are parametrized by 4-punctured rectangular
tori;  they and their conjugates are embedded.  We therefore
suggest the associate family morphing, and also morphing of
the modulus (aa) of the rectangular Torus, which changes the size
of the visible holes.

  For the visual appearance of these surfaces it is particularly
important that the punctures are centers of polar coordinate lines.
Formulas are taken from [K1] or [K2]. The Gauss maps for these
surfaces are degree 2 elliptic functions. The cases shown are
particularly symmetric, the zeros and poles of the Gauss map
are half-period points and the punctures are there. In the Jd-case
the diagonal of the rectangular fundamental domain joins the two
zeros, and in the Je-case it joins a zero and a pole of the Gauss map.

  Under suitable choices of the modulus of the Torus these
surfaces look like a fence of Scherk SaddleTowers - with
a vertical straight line (Je),  respectively a planar symmetry
line (Jd),  separating these towers. The conjugate surfaces
look qualitatively the same in the Jd-cases and  like a
checkerboard array of horizontal handles between vertical
planes in the Je-cases.

[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

Karcher_Je_Saddle_Tower.pdf