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This thesis is concerned with applications of Hodge theory in Teichmüller dynamics. Recall that the moduli space pairs (X, ω) of Riemann surfaces with a holomorphic 1-form carries a natural action of the group SL(2,R). The diagonal subgroup gives the Teichmüller geodesic flow, while full SL(2,R)-orbits give Teichmüller disks. The work of Eskin, Mirzakhani, and Mohammadi shows that the closure of a Teichmüller disk is always an immersed submanifold, usually called an “affine invariant submanifold” since it carries an affine structure. The first part of the thesis studies the Variation of Hodge Structures (VHS) over an affine manifold, and more generally over a Teichmüller disk. The affine manifold carries a finite measure and this allows one to extend many of the results in the ordinary theory of VHS to this setting. The second part of the thesis studies the Variation of Mixed Hodge Structures that arises in this setting. It shows that a certain part of it is particularly simple - it is split. This, in turn, allows for an algebraic characterization of affine manifolds.


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