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Fermi surfaces are surfaces of constant energy in reciprocal space. They provide information about the properties of a material in solid state physics.

Constant mean curvature surfaces are a superset of minimal surfaces, which minimize area and have zero mean curvature. Soap bubbles are surfaces of constant mean curvature.

The two certainly coincide in the case of a sphere, but must they always coincide? If so, why? Is there a sharper constraint that Fermi surfaces must obey as well?

I found a few papers that mention the two as connected, but they never establish the exact connection between the two.

Preliminary literature review:

Mackay, Alan L. Periodic minimal surfaces, 1985.

Minimal surfaces are found, as mentioned above, in soap films . . . The surfaces of constant energy in reciprocal space, used in solid state physics for finding the Fermi surface, are very similar.

Mackay, Alan L. Periodic minimal surfaces from finite element methods, 1994.

Fermi surfaces, which are surfaces in reciprocal space, are closely related to nodal surfaces.

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    $\begingroup$ If you had a charged fluid that formed bubbles and subjected it to a nonhomogogenous electric field, I doubt the bubbles would form surfaces of constant mean curvature, but I bet they'd still be Fermi surfaces. This isn't my field of expertise, though. $\endgroup$
    – Jerry Schirmer
    Sep 14, 2013 at 19:57
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    $\begingroup$ The (local) radius of curvature of a Fermi surface gives the effective mass at that location. The mean curvature would then give the mean effective mass for the two principal axes. There are many scenarios where the effective mass fails to be defined, such as at band crossings (like in graphene), so the very minimal condition for a constant mean curvature surface is having a single band Fermi surface. $\endgroup$
    – KF Gauss
    Mar 23, 2019 at 8:27

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"...must they always coincide?"

I think the answer is "no". The Fermi surface of graphene (under certain assumptions outlined in Section II A. of this paper) consists of 6 separate points. Locally, the energy as a function of k scales as |k|, so the points are not only isolated, they each form the sharp tip of a cone. Needless to say, curvature is not well defined for them.

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I can't say if they always coincide if pathological examples like graphene are excluded, but at least this is a partial answer.

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