  We classify the possible closures of leaves of the isoperiodic foliation
(sometimes called absolute period foliation) defined on the Hodge bundle, i.e.
the moduli space of abelian differentials over genus $g\geq 2$ smooth curves,
and prove that the foliation is ergodic on those sets. The results derive from
the connectedness properties of the fibers of the period map defined on the
Torelli cover of the moduli space. Some consequences on the topology of Hurwitz
spaces of primitive branched coverings over elliptic curves are also obtained.
To prove the results we develop the theory of augmented Torelli space, the
branched Torelli cover of the Deligne-Mumford compactification of the moduli
space of curves.
