  In this paper, we prove that extremal black holes arise on the threshold of
gravitational collapse. More precisely, we construct smooth one-parameter
families of smooth, spherically symmetric solutions to the
Einstein-Maxwell-Vlasov system which interpolate between dispersion and
collapse and for which the critical solution is an extremal black hole.
Physically, these solutions can be understood as beams of gravitationally
self-interacting collisionless charged particles fired into Minkowski space
from past infinity. Depending on the precise value of the parameter, we show
that the Vlasov matter either disperses due to the combined effects of angular
momentum and electromagnetic repulsion, or undergoes gravitational collapse. At
the critical value of the parameter, an extremal Reissner-Nordstr\"om black
hole is formed. No naked singularities occur as the extremal threshold is
crossed. We call this critical phenomenon extremal critical collapse and the
present work constitutes the first rigorous result on the black hole formation
threshold in general relativity.
