# the maths behind the performance

A Minimal Surface is a surface that locally minimises its area.

Minimal surfaces occur in nature, for example withdrawing a wire loop from soap water, the resulting soap film will minimise the area with the given boundary.

information presented sits within the context of research undertaken by Dr.Leschke at the University of Leicester through the international research programme m:iv.

Topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretchingtwistingcrumpling and bending, but not tearing or gluing. Minimal surface* study falls within this area.

What mathematicians call a minimal surface is not exactly what occurs in nature. A surface in mathematics doesn't have "thickness" nor does a minimal surface minimize area*. However if we zoom in to a point of the surface we find that it locally does.

Therefore, we are looking at a point of the surface that is enclosed in a curved boundary and which occupies the least area possible.

A minimal surface has vanishing average curvature ("vanishing mean curvature"): taking the curvature of all curves through a point, the average of the largest/smallest curvatures should vanish: each point looks like a saddle*.

Surfaces may have holes.

The number of holes is called the genus (g) of the surface. A sphere for example has genus zero, a torus has genus 1.

Mathematicians observe the properties of the 'zoomed in' points and come to a conclusion about the nature of the entire surface. If the constituent points of the surface minimise area then that is a minimal surface.

Minimal surfaces exists in lots of shapes and forms. To better understand these complex concepts, simplify and narrow down the area of study, mathematicians may choose to only focus on one type of minimal surface, here: the embedded surface

For a surface to be consider embedded it has to fulfil two preconditions. If we extend the edges of the shape out to infinity these should never cross each other (no self intersections) and should never come into contact (no touching points).

We want our surfaces to have no boundary as this enforces the surface to go out to infinity. For the purpose of this research, the edges or 'ends' (r) have to behave nicely. That means that as they extend into infinity the either look like a plane* or a catenoid*. That way they will also never intersect or touch!

To study existing and discover new minimal surfaces mathematicians use a variety of tools including complex maths and computer software. Their research involves deforming known minimal surfaces to discover new ones that might be able to maintain the properties mentioned above

*surface: In mathematics, a surface is a geometrical shape that resembles a deformed plane

*area: the space occupied by a flat shape or the surface of an object

*saddle point: where mean curvature is 0

*plane: a flat, two-dimensional surface that extends infinitely far

*catenoid: a type of surface, arising by rotating a catenary curve about an axis

The Finite Topology Conjecture

An orientable surface M of finite topology with genus g and r ends, r≠ 0, 2, occurs as a topological type of surface if and only if r ≤ g + 2