Riemann's Surface

riemann morph 001
riemann morph

The Gauss maps of these famous surfaces are the similarly famous Weierstrass p-functions on rectangular tori. [see Elliptic Function: JE]

Compare the image with the image of the Lopez-Ros No-Go Theorem Lopez-Ros-No-Go surface.

riemann associate 001
The Catenoid-like handles on one end of the animation are deformed to Helicoid-Catenoid helicoid like portions on the conjugate surface, see both sides of the rather squeezed handle of the conjugate surface.
riemann st
riemann st
riemann sw
riemann sw
       About Riemann's Minimal Surfaces

                      H. Karcher

   This is the family of singly-periodic embedded minimal
surfaces found by Riemann. They are parametrized (aa) by
rectangular tori. The Gauss map is the Weierstrass pe
function additively normalized to have a double zero at
the branch point diagonally opposite the double pole and
multiplicatively normalized to have the values plus or
minus i  at the four midpoints (on the Torus) between the
zero and the pole. The minimal surface has rotational
symmetries around the corresponding normals. This
symmetry kills the horizontal periods.  The surface is
parametrized by the range of the Gauss map with polar
coordinates around the punctures.
The surfaces look like families of parallel planes with one handle
between adjacent planes. The associate family morphing joins
two such embedded surfaces - they are congruent for the square
Torus. The standard morphing (aa) changes the branch values of
the Gauss map, i.e. the tilt of the normal at the flat points (K=0).

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].