Minimal surface

Minimal surface
For minimal surfaces in algebraic geometry see Minimal model (birational geometry).
Verrill Minimal Surface
Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature 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, which is a minimal surface whose boundary is the wire frame.

Contents

Examples

Classical examples of minimal surfaces include:

  • the plane, which is a trivial case
  • catenoids: minimal surfaces made by rotating a catenary once around its symmetry axis
  • helicoids: 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

Recent work in minimal surfaces has identified new completely embedded minimal surfaces, that is minimal surfaces which do not intersect. In particular Costa's minimal surface was first described mathematically in 1982 by Celso Costa and later visualized by Jim Hoffman. This was the first such surface to be discovered in over a hundred years. Jim Hoffman, David Hoffman and William Meeks III, then extended the definition to produce a family of surfaces with different rotational symmetries.

Minimal surfaces have become an area of intense mathematical and scientific study over the past 15 years, specifically in the areas of molecular engineering and materials science, due to their anticipated nanotechnology applications.

In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927- ), Robert Longhurst (1949 - ), Charles O. Perry (1929 - 2011), among others.

Definition

Given an embedded surface, or more generally an immersed surface (which may have a fixed boundary, possibly at infinity), one can define its mean curvature, and a minimal surface is one for which the mean curvature vanishes.

The term "minimal surface" is because these surfaces originally arose as surfaces that minimized surface area, subject to some constraint, such as total volume enclosed or a specified boundary, but the term is used more generally.

Minimal surfaces are the critical points for the mean curvature flow: these are both characterized as surfaces with vanishing mean curvature.

The definition of minimal surfaces can be generalized/extended to cover constant mean curvature surfaces: surfaces with a constant mean curvature, which need not equal zero.

A relation to Brownian motion

Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces. See R. Neel 2008 or arXiv version.

See also

References

External links


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