Isoperiodic meromorphic forms: two simple poles

Abstract: In this paper we prove that isoperiodic moduli spaces of meromorphic differentials with two simple poles on homologically marked smooth curves are non empty and connected, unless they correspond to double covers of $\mathbb{C}/\mathbb{Z}$ on curves of genus at least two. We deduce dynamical consequences for the corresponding isoperiodic foliation.