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I am currently interested in the differential geometry of minimal surfaces, and I have a rather trivial question regarding Scherk's surface (the one which can be parametrised by the real function $(x,y)\mapsto\ln\big(\frac{\cos(x)}{\cos(y)}\big)$).

On the internet, it appears that a fraction of this surface can be constructed/modelled with a cube in which some edges are removed, shrinking it into a soapy solution, as in the picture that I share with you just below.

enter image description here

However, how can one be completely certain that this really represents such a surface, and it is not just another surface that happens to look the same as Scherk's one?

I don't have enough bibliographical information to check whether someone already pointed out the link between the surface and such a model, so I would really appreciate some kind or reference so I can learn more about this, and other soap-film models.

Thank you in advance for your answers.

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  • $\begingroup$ I suppose it depends on what standard of argument you want. I think the standard one is that 1) soap films tend to form minimal surfaces (add usual caveats), 2) minimal surfaces with given boundary conditions are unique (again some caveats) and 3) the Scherk surface has the "square" boundary conditions. Which of these is your question about? $\endgroup$
    – Ryan Budney
    Commented Apr 16 at 16:11
  • $\begingroup$ Another approach is you could form the soap bubble and then measure its shape with some sort of lidar scanner, to compare with your given parametrization. I'm not aware of anyone doing that, but perhaps someone in solid-state physics or continuum mechanics has done it. That would be more of a question to ask to physicists or engineers. $\endgroup$
    – Ryan Budney
    Commented Apr 16 at 16:15
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    $\begingroup$ I would imagine your picture is not exactly the Scherk surface since gravity and air currents are distorting it a little. But modulo that small distortion it should be close to an exact match. $\endgroup$
    – Ryan Budney
    Commented Apr 16 at 16:20
  • $\begingroup$ Yeah, my question regards the boundary condition for the Scherk surface. $\endgroup$
    – Akerbeltz
    Commented Apr 16 at 17:37
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    $\begingroup$ For a certain idealization, the connection between soap-films and minimal surfaces was essentially known to Thomas Young and Pierre-Simon Laplace in the early 1800s (their study is technically for surface tension of a liquid, but the same derivation works for soap films too). $\endgroup$
    – Willie Wong
    Commented Apr 17 at 1:34

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Your question is actually related to a somewhat non-trivial property of the minimal graph equation (and related non-linear elliptic equations) that goes back to work of Jenkins and Serrin.

The basic idea is that the Scherk solution should be an example of a larger class of solutions considered by Jenkins-Serrin

In their paper, polygons (with an even number of faces) in the plane are considered and a type of infinite boundary condition for the minimal graph equation is imposed. This consists of asking that the function become $+\infty$ and $-\infty$ on alternating faces of the polygon. Under appropriate conditions on the polygon such a solution is showed to exist and is unique in the class of graphical solutions. The Scherk solution you wrote down is the Jenkin-Serrin solution for the square $[-\frac{\pi}{2}, \frac{\pi}{2}]\times [-\frac{\pi}{2}, \frac{\pi}{2}]$.

This relates to your question as this solution is constructed as the limit of the solutions of the minimal graph equation, $u_N$, that have finite boundary $+N$ and $-N$ on alternating faces. Such solutions were shown to exist earlier by work of Nitsche.

The minimal surface in your picture should (for physical reasons mentioned already) be well approximated by one of the $u_N$ and so should approximately what the Scherk surface looks like (am not sure how high the $N$ has to be to give an approximation to desired accuracy, but imagine it can be estimated).

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