Global properties of some weight 3 variations of Hodge structure

Abstract

We survey results on the global geometry of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). Included are uniformization results of domains in flag manifolds, a strong Torelli theorem, as well as the formula for the sum of Lyapunov exponents conjectured by Eskin, Kontsevich, Möller, and Zorich. Additionally, we establish the Anosov property of the monodromy representation, using gradient estimates of certain functions derived from the Hodge structure.