This paper describes the intrinsic geometry of a leaf of the absolute period foliation of the Hodge bundle : its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of for general and then turn to a study of the case . In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on whose zeros, poles, and exotic trajectories are analyzed in detail.
"Moduli spaces of isoperiodic forms on Riemann surfaces." Duke Math. J. 163 (12) 2271 - 2323, 15 September 2014. https://doi.org/10.1215/00127094-2785588