  Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in
suspended samples and rolling-up often occurs when boundaries are not fixed. We
address this variety of graphene geometries by classifying all ground-state
deformations of the hexagonal lattice with respect to configurational energies
including two- and three-body terms. As a consequence, we prove that all
ground-state deformations are either periodic in one direction, as in the case
of ripples, or rolled up, as in the case of nanotubes.
