Why is the catenoid the minimal surface of revolution?

Question

This is more of a soft question than anything and I'm asking for either a proof or intuitive explanation as to why this is.

It seems that if one defines 2 points in the upper-half of the Cartesian plane and one wishes to define a function between them that minimises the surface area of the surface-of-revolution, one should draw a straight line between the two points. This is the shortest distance between any points on the cartesian plane and so one would think that its associated surface-of-revolution would be the smallest in surface area - the catenoid, however, is supposedly even smaller!

Why is this the case? Why is the catenary a "better" function in terms of minimising surface area than the straight line?

Answer 1

Here are three arguments, all intuitive and hence not totally convincing.

First, a physical argument. Let's say we place two horizontal circles, made out of wire, at $z=0$ and $z=h$. Now we dip them in soapy water, to make a soap film. Like so:

(Image taken from Soap Film and Minimal Surface, which has a derivation of the catenoid.) Because of surface tension, the film tries to make its area as small as possible. (Strictly speaking we should do this in zero-gravity.) I hope it seems intuitively plausible that a cylinder is not what to expect for the soap film.

Second argument: minimal surfaces are characterized locally by having mean curvature equal to zero. The mean curvature is the average of the maximal and minimal (signed) principal curvatures. That is, you consider all planes perpendicular to the surface at a point, and look at the curvatures of the intersection curves. Consider a point at the "waist" of the surface. The two principal curves curve in opposite directions, giving a zero sum. But for the cylinder, one principal curve is a circle and the other is a straight line, so you get a nonzero mean curvature.

Third argument: let's say we're trying to decide how to draw a curve from $(r,0)$ to $(r,h)$ in the plane so that the surface of revolution around the $z$-axis will be as small as possible. In what direction should we depart from the lower point? If we set out vertically, it's true we make the length of the generating curve as small as possible. But if we head in towards the $z$-axis, we make the radii smaller, which tends to make the surface of revolution smaller in area. So it's a balancing act.

This is a problem in the calculus of variations. The Feynman Lectures (sec.II-19), "The Principle of Least Action", discusses a (mathematically) similar problem: the path of a projectile in a uniform gravitation field. As you'd expect, Feynman gives good intuition. Here we are minimizing the integral of (kinetic energy)-(potential energy) instead of surface area, but mathematically it looks much the same. Two key paragraphs:

Now, an object thrown up in a gravitational field does rise faster first and then slow down. That is because there is also the potential energy, and we must have the least difference of kinetic and potential energy on the average. Because the potential energy rises as we go up in space, we will get a lower difference if we can get as soon as possible up to where there is a high potential energy. Then we can take that potential away from the kinetic energy and get a lower average. So it is better to take a path which goes up and gets a lot of negative stuff from the potential energy [there's a figure].

On the other hand, you can’t go up too fast, or too far, because you will then have too much kinetic energy involved—you have to go very fast to get way up and come down again in the fixed amount of time available. So you don’t want to go too far up, but you want to go up some. So it turns out that the solution is some kind of balance between trying to get more potential energy with the least amount of extra kinetic energy—trying to get the difference, kinetic minus the potential, as small as possible.

The soap film is also minimizing energy---in this case just the total potential energy coming from the surface tension, which is proportional to the total area. So the two cases are rather similar.