Schwarz PD Family Surfaces

Differential Equations, Mechanics, and Computation
minimal surface
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       About the Schwarz PD Family

                     H. Karcher

  This is a 2-parameter family of triply periodic genus 3 surfaces.
In each case the original surface and the conjugate surface are
embedded. The most symmetric example (with a cubic lattice)
which is obtained when cc = 0, dd = 0,  was already 
constructed by H. A. Schwarz.  When Alan Schoen
found more triply periodic surfaces around 1970 he named the
two surfaces which Schwarz found the P-surface (P for cubic
primitive) and the D-surface (D for diamond). He also found a
third embedded(!) surface in the associate family of these, the
Gyroid (associate parameter 0.577 which is approx. 52 degrees).
If dd=0 then a fundamental cell for the lattice is a prisma with
square base, in the morphing cc changes the height of the prisma.

K. Grosse-Brauckmann, M. Wohlgemuth: The Gyroid is embedded
and has constant mean curvature companions.
To appear Calc. Var. 1996

  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
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