In differential geometry, Riemann's minimal surface is a one-parameter family of

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...

s described by Bernhard Riemann in a posthumous paper published in 1867. Surfaces in the family are singly periodic minimal surfaces with an infinite number of

ends End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games ** End (gridiron footbal ...

asymptotic to parallel planes, each plane "shelf" connected with

catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...

-like bridges to the neighbouring ones. Their intersections with horizontal planes are circles or lines; Riemann proved that they were the only minimal surfaces fibered by circles in parallel planes besides the

helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...

and plane. They are also the only nontrivial embedded minimal surfaces in Euclidean 3-space invariant under the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

generated by a nontrivial translation. It is possible to attach extra handles to the surfaces, producing higher-

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

minimal surface families.

References

External links

* http://www.math.indiana.edu/gallery/minimalSurface.phtml * http://www.indiana.edu/~minimal/essays/riemann/index.html * http://virtualmathmuseum.org/Surface/riemann/riemann.html {{Bernhard Riemann Differential geometry Minimal surfaces Bernhard Riemann