  We investigate the cosmology of mini Primordial Black Holes (PBHs) produced
by large density perturbations. The mini PBHs evaporate promptly in the early
universe and we assume that a stable remnant is left behind. The PBHs remnants
can constitute the entire dark matter of the universe for a wide range of
remnant masses. We build inflationary models, in the framework of
$\alpha$-attractors utilizing exponential functions, in which the PBHs are
produced during matter, radiation and kination domination eras. The advantage
of these inflationary models is that the spectral index takes values favorable
by the Planck 2018 data. The PBH production from runaway inflaton models has
the unique and very attractive feature to automatically reheat the universe. In
these models the PBHs are produced during the kination stage and their prompt
evaporation efficiently produces the required entropy.
  Such runaway models are remarkably economic having interesting implications
for the early universe cosmology, possibly giving rise to a wCDM late time
cosmology as well.
