  Samarium hexaboride (SmB$_6$), a representative Kondo insulator, has been
characterized recently as a likely topological insulator. It is also a material
with strong electron correlations, evident by the temperature dependence of its
bandgap and the existence of a nearly flat collective mode whose energy lies
within the bandgap. Similar strong correlations can affect or even destabilize
the two-dimensional metallic state of topological origin at the crystal
boundary. Here we construct the minimal lattice model of the correlated
boundary of topological Kondo insulators, and make phenomenological predictions
for its possible ground states. Depending on the microscopic properties of the
interface between the topological Kondo material and a conventional insulator,
the boundary metal can exhibit a varied degree of hybridization between the $d$
and $f$ orbitals of the rare earth element, yielding a rich two-dimensional
heavy fermion phenomenology. A pronounced participation of the $f$ orbitals is
expected to create a heavy fermion Dirac metal, possibly unstable to a spin
density wave, electron localization or even superconductivity. The opposite
limit of "localized magnetic moments" helped by the partial Kondo screening on
the crystal boundary can bring about a non-Fermi liquid of $d$ electrons that
exhibits two-dimensional quantum electrodynamics, or other unconventional
states. In addition, ultra-thin films made from topological Kondo insulators
could open the possibility of creating exotic incompressible quantum liquids
with non-Abelian fractional excitations, whose dynamics shaped by the strong
Rashba spin-orbit coupling resembles that of fractional quantum Hall systems.
