  Twisting van der Waals heterostructures to induce correlated many-body states
provides a novel tuning mechanism in solid-state physics. In this work, we
theoretically investigate the fate of the surface Dirac cone of a
three-dimensional topological insulator subject to a superlattice potential.
Using a combination of diagrammatic perturbation theory, lattice model
simulations, and ab initio calculations we elucidate the unique aspects of
twisting a single Dirac cone with an induced moir\'e potential and the role of
the bulk topology on the reconstructed surface band structure. We report a
dramatic renormalization of the surface Dirac cone velocity as well as
demonstrate a topological obstruction to the formation of isolated minibands.
Due to the topological nature of the bulk, surface band gaps cannot open;
instead, additional satellite Dirac cones emerge, which can be highly
anisotropic and made quite flat. We discuss the implications of our findings
for future experiments.
