Moduli spaces of isoperiodic forms on Riemann surfaces

Abstract

This paper describes the intrinsic geometry of a leaf $\mathcal{A}\left(L\right)$ of the absolute period foliation of the Hodge bundle $\Omega {\overline{\mathcal{M}}}_g$: its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of $\mathcal{A}\left(L\right)$ for general $g$ and then turn to a study of the case $g=2$. In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on $\mathcal{A}\left(L\right)\cong \mathbb{H}$ whose zeros, poles, and exotic trajectories are analyzed in detail.