7.3: Minimal Surfaces and Their Deformations

Recall that the mean curvature H(u,v) of a surface is given by the equation

Xu(u,v) × Nv(u,v) + Nu(u,v) × Xv(u,v) = 2H(u,v)Xu × Xv.

A minimal surface is defined by the condition that the mean curvature H(u,v) is zero at all points of the surface.

Mean Curvature for Surfaces

This demonstration is similar to the various demos on Gaussian curvature in that there is a square patch in the Domain window that can be moved around and resized using the two white dots (hotspots). The patch is color-coded and its image apears on the surface. The third window displays a graph of the mean curvature of the surface at each point in the domain. The graph of H(u,v) is clored using a gradient so that graph is blue whenever H(u,v) < 0, white whenever H(u,v) = 0, and red whenever H(u,v) > 0. This coloring scheme makes it easy to tell whether a surface is minimal.

In the following paragraphs, we discuss a number of examples of minimal surfaces. You are encouraged to work with these examples and corresponding exercises and to refer to the above demonstration.

The Enneper Surface

The following surface was discovered by Enneper: X(u,v) = (3u(1+v2) - u3, 3v(1+u2) -v3, 3c(u2-v2))
 For which domains will this surface intersect itself? For which values of c will the Enneper surface be minimal?

Exercises

(Observe the mean curvature graphs for the following surfaces and find the values of the constant c such that the surface is minimal if there are any such values)

    1. This is a more general surface of Enneper type:
      X(u,v) =&parl3;ucos(v) - u2c+1cos((2c+1)v)2c+1 ,
      usin(v) -u2c+1sin(( 2c+1)v)2c+1 ,
      2u2c+1cos((2c+1)v)2c+1&parr3;

    2. The following surface is named for its discoverer, Scherk. The domain is the "checkerboard", where cos(u) and cos(v) have the same algebraic sign. It is parametrized by the following:
      X(u,v) = (u,v,ln(cos(v)/cos(u)))

    3. The following is a sequence of special catenoid surfaces:
      X(u,v) = (1/c)cosh(2v)cos(u), (1/c)cosh(2v)sin(u), v)
    For which values of c will the surface be minimal?

For minimal surfaces in three-space, it is always possible to find a one-parameter deformation such that all surfaces in the family are minimal and all have the same normal mapping. We can see this in the most famous of all such deformations, from the catenoid to the helicoid.

Deformations of Minimal Surfaces

Consider the following family of surfaces, where the parameter c ranges from 0 to π/2

Let

X(u,v) = (cosh(v)cos(u),cosh(v)sin(u),v)

Y(u,v) = (sinh(v)sin(u),-sinh(v)cos(u),u)

Now consider the mean curvature, the Gaussian curvature, and the normal mapping of cos(c)X(u,v)+sin(c)Y(u,v). The following demo animates the deformation from the catenoid to the helicoid, using this equation, and displays the graph of mean curvature.

Exercises

    1. What in particular happens to the graph of Gaussian curvature of the deformed surface? How about the mean curvature?

    2. Find the     Lij matrices for the catenoid and the helicoid. Calculate the determinant and trace of each matrix and use your results to check your observations about the Gaussian and mean curvatures from the previous question.

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