  Superlattice potential modulation can produce flat minibands in
Bernal-stacked bilayer graphene. In this work we study how band topology and
interaction-induced symmetry-broken phases in this system are controlled by
tuning the displacement field and the shape and strength of the superlattice
potential. We use an analytic perturbative analysis to demonstrate that
topological flat bands are favored by a honeycomb-lattice-shaped potential, and
numerics to show that the robustness of topological bands depends on both the
displacement field strength and the periodicity of the superlattice potential.
At integer fillings of the topological flat bands, the strength of the
displacement field and the superlattice potential tune phase transitions
between quantum anomalous Hall insulator, trivial insulator, and metallic
states. We present mean-field phase diagrams in a gate voltage parameter space
at filling factor $\nu=1$, and discuss the prospects of realizing quantum
anomalous Hall insulators and fractional Chern insulators when the superlattice
potential modulation is produced by dielectric patterning or adjacent moir\'e
materials.
