Why is the minimal surface of revolution not a cylinder?

Question

I'm trying to find the curve $r=f(z)$ that links two parallel circles of same radii, aligned on the same axis $ (Oz) $, and that minimizes the surface of revolution between both circles.

The solution seems to be a catenoid :

However, I can't wrap my head around the fact that it's not just the cylinder connecting both circles :

Intuitively, I'd want to minimize the length of the curve connecting two vertically aligned points of the corcle, an then totate that curve around the axis. Thus, that curve would be the straight segment from one circle to the other, parallel to $ (Oz) $.

The surface we need to minimize seems to be equal to : $S= l \times 2 \pi R $ , with $ l $ the length of the curve between both circles and $ R $ the radius of both circles.

Thus, I'd want to minimize $l$ : I' would simply get a straight line between both circles.

I saw this same question asked on another post, and the answers seem to revolve around changing the radius of the curve along it's path between both circles. However, I still can't grasp completely why the solution ins't just the cylinder. Intuitively, I agree with the "soap bubble" example, but I can't reconcile that physical intuition with the idea that it actually, mathematically, minimizes the surface area between both circles.

Simply said, if I were to paint the exterior of both of these shapes, it seems to me that I would need more paint to cover the catenoid than the cylinder, hence a greater surface area?

Am I understanding the "surface" this problem refers to correctly ?

Any help in understanding this would be greaty appreciated, thanks.

Answer 1

Inuitively, each very very thin (i.e., of width $dw$) band between the two circles contributes a total of $\frac{2\pi r}{\cos\theta}dw$ towards the whole surface area, where

• $r$ is the radius of the "cylinder" at the band location, i.e. the radius of the band itself
• $\theta$ is the angle of the surface (away from the vertical)

So,

• if the top of our surface is always vertical, then we get $2\pi r$ from each band
• However, if the surface angles towards the center, then sure, we get a little more surface from the initial bands near the circles, but the bands near the center can have a substantially smaller $r$, such that the decrease in their contributions more than outweighs the slighly increased surface near the top and bottom.