A transfer principle: from periods to isoperiodic foliations

We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus g >= 2 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.