Moduli Spaces of Isoperiodic Forms on Riemann Surfaces

Abstract

This paper describes the intrinsic geometry of a leaf (\mathcal{A}(L)) of the absolute period foliation of the Hodge bundle (\Omega \bar{M}_g): its singular Euclidean structure, its natural foliations and its discretized Teichmuller dynamics. We establish metric completeness of (\mathcal{A}(L)) 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}(L) \cong \mathbb{H}), whose zeros, poles and exotic trajectories are analyzed in detail.