  Twisted graphene multilayers have demonstrated to yield a versatile
playground to engineer controllable electronic states. Here, by combining
first-principles calculations and low-energy models, we demonstrate that
twisted graphene trilayers provide a tunable system where van Hove
singularities can be controlled electrically. In particular, it is shown that
besides the band flattening, bulk valley currents appear, which can be quenched
by local chemical dopants. We finally show that in the presence of electronic
interactions, a non-uniform superfluid density emerges, whose non-uniformity
gives rise to spectroscopic signatures in dispersive higher energy bands. Our
results put forward twisted trilayers as a tunable van der Waals
heterostructure displaying electrically controllable flat bands and bulk valley
currents.
