minimal surface between two long distance aligned circles

Question

Here is my problem: If I take two circles , let's say (cos(t), sin(t),0) and (cos(t), sin(t),x) (t between 0 and 2pi), when x is small the minimale surface between them looks like a slightly incurved cylinder. Here I call "minimale surface" a smooth (non necessarily connected) surface S of minimale area such that the boundary of S is exactly the two circles. My question is about what happens when x goes to infinity: Is there a point where "the" minimal surface becomes the union of two disks, or does it always stay diffeomorphic to a cylinder ? Thanks!

Answer 1

Actually, the maximum plane separation is about $2/3$ of the diameter when the two circles have the same diameter. The exact figure near $2/3$ is the reciprocal of the slope of the tangent line to the standard catenoid that passes through the origin. From my calculator, the slope is $s \approx 1.508879561,$ so the reciprocal is $\frac{1}{s} \approx 0.6627434$

There is another way to discuss this, published by Nitsche, but there is a quick proof. It is not possible to have the planes separated so far apart (say, more than that diameter) that two copies of the surface, one rotated $90^\circ,$ can be separated laterally and then brought into tangential contact in an interior point. This is prohibited by the maximum principle

There is a surprising part: if we misalign the circles a little bit, there will still be a sort of tube minimal surface joining them, it will just be a bit slanted rather than being a surface of revolution anymore. The surprise, due to Riemann, is that the intesection of this tube with any parallel plane is also a circle. There is no single agreed upon name for the resulting surface, i like to call it the Riemann Staircase, because the phrase Riemann Surface already has too many important associations.