  We explore the use of applied electrical potentials to induce band flattening
in graphene for bands near zero energy. We consider various families of doubly
periodic potentials and simulate their effect on the electronic band structure
using a tight-binding and a continuum approach. From these families, we find
that an applied potential with symmetries of wallpaper group 17, in particular
a Kagome potential, works best for inducing a high degree of band flattening
for a range of realistic potential amplitudes and periods. Our work indicates
that it should be possible to engineer the band structure of graphene using
electrostatic gating, thus enabling a new approach to the development of
graphene-based metamaterials.
