Enneper's Minimal Surface

Date: 10/22/2001 at 22:49:59
From: Robert Vermillion
Subject: Enneper Minimal Surface

What is an Enneper Minimal Surface? Everywhere that I've seen talks 
about what they look like, but so far I have not located any plans for 
generating an EMS. Why is it minimal as opposed to maximum? What 
quantity of the surface is minimal or how can one recognize a given 
curve as an EMS?  Could you assist?  

Thank you kindly, sir.

Date: 10/23/2001 at 00:18:32
From: Doctor Pete
Subject: Re: Enneper Minimal Surface

Hi,

Enneper's minimal surface is defined in 3-dimensional Cartesian 
coordinates by the equations

     x = u - (u^3)/3 + uv^2
     y = v - (v^3)/3 + vu^2
     z = u^2 - v^2,

for real u, v. That is, the point (u,v) is mapped to 
(x(u,v),y(u,v),z(u,v)), which is a point on Enneper's minimal surface. 
While this surface has self-intersections in 3-space, it is minimal 
because it has the property of minimizing surface area. This notion 
can be made more precise by saying that the surface satisfies a set of 
differential equations, but another way of describing minimal surfaces 
is that their mean curvatures at every point is zero.

To understand the concept of mean curvature, we begin by describing 
the curvatures of a surface at a point. Consider a vector normal to 
the surface at the given point; this vector is perpendicular to the 
tangent plane at that point. If you look at the 2-d curve formed by 
the intersection of any plane containing the normal vector, it has a 
curvature at the given point. Depending on the particular plane you 
choose, the curvature may be larger or smaller; the principal 
curvatures are the maximum and minimum curvatures thus obtainable.  
The mean curvature is the arithmetic mean (average) of the two 
principal curvatures. Geometrically speaking, the surface is
"saddle-shaped" at that point. If, at every point on the surface, the 
mean curvature is zero, then the surface is minimal.

Minimal surfaces are sometimes called "soap-bubble films," in the 
sense that these surfaces minimize their surface area for a given 
boundary. If you imagine a loop of wire shaped into a circle, then the 
minimal surface that spans the wire is a disk, which is a subset of 
the plane, which is a minimal surface. If you take two such loops, 
parallel to each other, and dip them into a soap solution, you may 
obtain a catenoid, which is another example of a minimal surface.  
Other embedded minimal surfaces include helicoids, Scherk surfaces, 
and more recently the Costa-Hoffman-Meeks family of minimal surfaces.

If you're interested in minimal surfaces, I recommend a textbook on
differential geometry. A background in multivariable calculus would be
essential in order to discuss relevant concepts which include 
curvature of a surface at a point, the Gauss map and global curvature.

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/23/2001 at 02:17:42
From: Robert Vermillion
Subject: Re: Enneper Minimal Surface

Thanks for shedding informative light upon the hitherto dark subject 
of the Enneper Minimal Surface. Your explanation was pleasantly 
concise, erudite, comprehensible, and satisfyingly explanatory. When I 
have another such question, I'll come to the same place.  :)
                                                                              
Robert Vermillion