FINITE RIEMANN'S MINIMAL SURFACE

 Surface studied by David Hoffman and Hermann Karcher in 1993. Links:  www.indiana.edu/~minimal/archive/Spheres/Planar/FiniteRiemann/web/index.html www.indiana.edu/~minimal/archive/Harmonic/harmonic/(1,2,2)x2+(0,2,2)/web/index.html http://virtualmathmuseum.org/Surface/lopez-ros/lopez-ros.html www.indiana.edu/~minimal/research/claynotes.pdf p 29

The Riemann finite minimal surface is the minimal surface obtained by taking  (and then ) in the Weierstrass parametrization of a minimal surface.
It has two infinite sheets of the type of those of the catenoid and an infinite sheet with an asymptote plane (hence the name "finite", as opposed to Riemann's original minimal surface that has an infinite number of plane sheets).

Note that if extended, the sheets intersect, contrary to those of Costa's surface. See here a surface of the same type, non minimal but harmonic, the sheets of which do not intersect.

 The Riemann finite minimal surface has the same shape as the cubic surfaces with equations  and  .... asymptote cylinder x²+y²=a² asymptote cone x²+y²=z²
 ... or as the sextic surface with equation , as well as the transcendental surface with equation . For both of them, the asymptote lines are the straight lines .

 A Riemann finite surface, by Alain Esculier. A Riemann finite surface in wire, by Christoph Soland, Bugnon gymnasium, Lausanne, see at Dyck's surface.

Engraving of this surface, by Patrice Jeener