Whilst working on this problem, I've come across the work of Konrad Polthier on generalizing the notion of curature to discrete surfaces; much of his work is covered in his thesis, "Polyhedral surfaces of constant mean curvature". However, I have very little background knowledge of curvature and am struggling to follow much of what is being said.
I want to work with polyhedral surfaces more general than simplicial complexes (i.e. having possibly non-triangular faces), so I can no longer move the vertices around independently. Polthier mentions in passing that mean curvature can be defined on the edges of a surface rather than its vertices; this presumably corresponds to the fact that the realization space of a polyhedral graph has one degree of freedom for each edge, a conclusion which I reached as follows:
Intuitively, we can move any face of a polyhedron by defining a linear normal vector field on the plane in which it lies; this is equivalent to choosing three points in the face, and displacing them each by some amount in the normal direction. If the displacement is small and the polyhedron is "simple" (three faces meeting at each vertex), the new polyhedron will be combinatorially the same and the degrees of freedom are 3F = E + 6. However, having an extra face meeting at a vertex corresponds to removing an edge by contracting it to a point, and reduces the degrees of freedom by 1 if we want the new polyhedron to remain combinatorially the same; so the degrees of freedom are E + 6 in general. Finally, six of these degrees of freedom correspond to scaling, rotation and translation, i.e. similarity-preserving transformations. So the "shape" only really depends on E values.
Polthier defines mean curvature on edges in terms of their length and dihedral angle, but it's not clear to me whether this curvature can be varied independently on each edge in a canonical way (or even at all!) in the case of general polyhedral surfaces. The fact that there's one degree of freedom for each edge even in the general case makes me feel that there should be some natural correspondence, but I can't see what it is...
I tried explicitly working out the change in area which would result from moving a single face of a polyhedron in the aforementioned way, with the hope of deriving mean curvature as the gradient, but I failed because I'm shockingly bad at doing geometry with vectors.
Ultimately, I'm hoping to connect the "missing" degrees of freedom in non-simple polyhedra with the conjecture that they should never be global optimizers of surface area for a fixed number of faces; and, hopefully, show that the global optimizer (which is proven to exist) will never have faces with a large number of edges, because these would have large internal angles and small dihedral angles on their edges.
Sorry if any of this is vaguely worded. I'd greatly appreciate any help!
