Soap Film

I use the term soap film to mean a surface that does not exclose a volume, the way a soap bubble does. A soap film forms a minimal surface, which means that any small deformation of the surface would have more area. The mean curvature of a surface is the average of the two prinicpal curvatures. Every minimal surface has zero mean curvature. The converse, however, is not true. In other words, some surfaces with zero mean curvature are not minimal.

There is a useful analogy to functions of two variables: A minimal surface corresponds to a local minimum of a function of two variables, and a surface of zero mean curvature corresponds to a critical point (a local minimum or a saddle point). Analogous to "every minimal surface has zero mean curvature" is the fact that every local minimum of a function of two variables is a critical point. The converse is not true: not all critical points are local minima. For example, f(x,y) = x² - y² has a critical point at (x, y) = (0, 0), but the origin is a saddle point, not a local minimum.

Note: My definition of "minimal surface" is nonstandard. In many textbooks a minimal surface is defined to be one with zero mean curvature. Many of the exotic "minimal surfaces" you may read about or see on the DPGraph gallery are unstable, because they are not minimal surfaces as I define them. My minimal surface is a local minimum of the area functional, whereas a surface with zero mean curvature is a critical point of the area functional.

Theorem: A surface of revolution has zero mean curvature if and only if it is either a part of a plane or a part of a catenoid described in polar coordinates by

Thus, every surface of revolution with r = c cosh(z/c) has zero mean curvature, but sometimes the surface is not minimal since a perturbation of the surface leads to smaller area. How is this possible? Assume that the rings have radius 1, and that they are placed at z = ± z*. Then z* and c are related by

The unstable catenoids on the red curve are saddle points of the area functional: most perturbations of the surface make the area larger but one perturbation makes the area smaller. For any perturbation, the change in area is proportioal to the square of the strength of the perturbation (this is a consequence of the zero mean curvature of the surface.)

This is an example of a fold catastrophe (or a saddle-node bifurcation) in an infinite dimensional system. As the distance between the rings (2 z*) increases, a stable and unstable solution coalesce and annihilate each other.

The following figure plots z* = c acosh(1/c) in black, and r = c cosh(z/c) for c = 0.5524 in red. The critical soap film is the red curve, reflected across the r-axis and rotated about the z-axis. If c is larger than 0.5524 the soap film is stable, if c is smaller the soap film is unstable. Click on the figure to see a DPGraph version of the figure where c can be changed with the scrollbar. The value of c is the r-intercept of the figure, since r = c at z = 0.

Click here for a DPGraph of the soap film where c is constant in time, but can be changed with the scrollbar. The initial value of c is c = 0.5524 which gives the critical soap film. You can change c to the stable or unstable region, whereas the animation only shows stable soap films.

The following figure shows the area of the two catenoids at each value of z* < 0.6627, along with the area of two disks of radius 1 (which is another minimal solution.)

The next figure shows the area as a function of a for two values of z*. The black dots correspond to the soap film forming two disks with total area 2p, which is stable. The red dot corresponds to an unstable soap film, and the blue dot corresponds to a stable soap film. See the dots in the previous figure.

   

DPGraph implementation note:
In the "popping" soap film, c varies with time from .5525+.5 = 1.0525 down to .5525 in a saw-tooth manner. (That is, c is piecewise linear with jumps from .5525 up to 1.0525.) When c > 1, then acos(1/c) is undefined and there is no surface plotted. This makes the surface appear to "pop" when c jumps up to 1.0525.

References:
"Elementary Differential Geometry," by Barrett O'Neill proves the theorem stated here. This book is a good place to read about "zero mean curvature."

"The Mathematics of Soap Films: Explorations with Maple," by John Oprea. This is a charming little book. The current web site is a rediscovery of section 5.6, entitled "The Catenoid versus Two Disks," of Oprea's book.

"Soap Bubbles: Their Colors and Forces Which Mold Them," by C. V. Boys is a classic. The 1911 edition is reprinted and available from Dover.

These graphs were produced by DPGraph, which is a fast program for viewing 3D objects, and Mathematica.