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The problems of minimizing the potential energy of electrons on a sphere, or maximizing the smallest distance between the electrons, have been well-studied. E.g., see the earlier MO question "Distributing points evenly on a sphere."

My question concerns the same problems on an ellipsoid with two axis dimensions $a=b$ equal, and the third dimension $c$ much shorter, so the ellipsoid is pancake-like:

          EllipsoidSaucer
          Ellipsoid axis dimensions: $a=5, b=5, c=1$, with $c$ the vertical dimension.
It is natural to expect that with $c \ll a,b$, the electrons form an approximate honeycomb hexagonal packing on the upper and lower surface of the ellipsoid, perhaps with some edge effects near the horizontal midplane.

Has this problem been studied? Are there results established under certain conditions?

Update. Here is a nice image from the Müller/Frauendiener paper that Carlo Beenakker cites:
          ElectronsTorus
          Fig.6 (detail): $1024$ charged particles on a torus.

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    $\begingroup$ Is the ellipsoid an isolator? Would be yet another variant under which the solutions may be different. $\endgroup$
    – Manfred Weis
    Commented Feb 28, 2016 at 14:32
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    $\begingroup$ The answer for minimizing the potential energy (I am assuming a Coulomb potential) and maximizing the minimum distance are going to be very different. The latter gives an asymptotically uniform distribution, while the former will give an uneven distribution with much lower density on the top and bottom. The transition between asymptotic uniformity and nonuniformity is discussed in Hardin and Saff, 2005 dx.doi.org/10.1016/j.aim.2004.05.006 $\endgroup$
    – Yoav Kallus
    Commented Feb 28, 2016 at 14:42
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    $\begingroup$ software that will calculate this for you is provided in Charged particles constrained to a curved surface $\endgroup$
    – Carlo Beenakker
    Commented Feb 28, 2016 at 14:42
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    $\begingroup$ A little Googling came up with this homework problem(?): (physics.princeton.edu/~mcdonald/examples/ellipsoid.pdf), but I haven't checked the calculation. The electrostatic calculation gives the asymptotic density when the number of electrons is very large. $\endgroup$
    – Yoav Kallus
    Commented Feb 28, 2016 at 14:54
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    $\begingroup$ The continuous case has been solved for a more general class of shapes that includes oblate ellipsoids. See I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016 and math.stackexchange.com/questions/112662/… . $\endgroup$
    – user21349
    Commented Feb 28, 2016 at 15:21

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