  The moir\'e superlattice formed by ABC stacked trilayer graphene aligned with
a hexagonal boron nitride substrate (TG/h-BN) provides an interesting system
where both the bandwidth and the topology can be tuned by an applied
perpendicular electric field $D$ . Thus the TG/h-BN system can simulate both
Hubbard model physics and nearly flat Chern band physics within one sample. We
derive lattice models for both signs of $D$ (which controls the band topology)
separately through explicit Wannier orbital construction and mapping of Coulomb
interaction. When the bands are topologically trivial, we discuss possible
candidates for Mott insulators at integer number of holes per site (labeled as
$\nu_T$). These include both broken symmetry states and quantum spin liquid
insulators which may be particularly favorable in the vicinity of the Mott
transition. We propose feasible experiments to study carefully the bandwidth
tuned and the doping tuned Mott metal-insulator transition at both $\nu_T=1$
and $\nu_T=2$. We discuss the interesting possibility of probing experimentally
a bandwidth (or doping) controlled continuous Mott transition between a Fermi
liquid metal and a quantum spin liquid insulator. Finally we also show that the
system has a large valley Zeeman coupling to a small out-of-plane magnetic
field, which can be used to control the valley degree of freedom.
