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.