Title: Bi-Lagrangian structures and Teichmüller theory,
Abstract: This paper has two purposes: the first is to study several structures on
manifolds in the general setting of real and complex differential geometry; the
second is to apply this study to Teichmüller theory. We primarily focus on
bi-Lagrangian structures, which are the data of a symplectic structure and a
pair of transverse Lagrangian foliations, and are equivalent to para-Kähler
structures. First we carefully study real and complex bi-Lagrangian structures
and discuss other closely related structures and their interrelationships. Next
we prove the existence of a canonical complex bi-Lagrangian structure in the
complexification of any real-analytic Kähler manifold and showcase its
properties. We later use this bi-Lagrangian structure to construct a natural
almost hyper-Hermitian structure. We then specialize our study to moduli spaces
of geometric structures on closed surfaces, which tend to have a rich
symplectic structure. We show that some of the recognized geometric features of
these moduli spaces are formal consequences of the general theory, while
revealing other new geometric features. We also gain clarity on several
well-known results of Teichmüller theory by deriving them from pure
differential geometric machinery.