Publications

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1) . We show … that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual boundary of hyperbolic 3-space. This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group G. These manifolds … are obtained as quotients of the domains of discontinuity in generalized flag varieties G/P constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For G-Fuchsian representations and their Anosov deformations, where G is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For G-Fuchsian and G-quasi-Fuchsian representations in simple G of rank at least two, we show that the quotient manifolds are not Kähler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for G of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a G-Fuchsian representation in \mathrmPSL_3(C) is a fiber bundle over a surface, and we conjecture that this holds for all simple G.

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.