In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a minimal surface is a surface with a mean curvature

Mean curvature

In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....

 of zero.
These include, but are not limited to, surfaces of minimum area subject to various constraints.

Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film

Soap film

Soap films are thin layers of liquid surrounded by air. For example, if two soap bubbles enters in contact, they merged and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders...

, which is a minimal surface whose boundary is the wire frame.
Classical examples of minimal surfaces include:

• the plane, which is a trivial
  Trivial (mathematics)
  In mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
  
   case
• catenoid
  Catenoid
  A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...
  
  s: minimal surfaces made by rotating a catenary
  Catenary
  In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...
  
   once around its symmetry axis
• helicoid
  Helicoid
  The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
  
  s: A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity
• the Enneper surface
  Enneper surface
  In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a surface that can be described parametrically by:It was introduced by Alfred Enneper in connection with minimal surface theory....
  
  

Recent work in minimal surfaces has identified new completely embedded minimal surfaces, that is minimal surfaces which do not intersect.