Ioannis D. Platis

Geometry of Riemann Surfaces: Complex hyperbolic quasi-Fuchsian groups

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current state of the art of the theory of complex hyperbolic quasi-Fuchsian groups.

Global Geometrical Coordinates on Falbel's Cross-Ratio Variety

Falbel has shown that four pairwise distinct points on the boundary of a complex hyper- bolic 2-space are completely determined, up to conjugation in PU(2,1), by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.

The geometry of complex hyperbolic packs

Complex hyperbolic packs are 3-hypersurfaces of complex hyperbolic plane H2C which may be considered as dual to the well known bisectors. In this article we study the geometric aspects associated to packs.

Complex Symplectic Geometry of Quasi-Fuchsian Space

We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmüller space and is described in terms which look natural from the point of view of hyperbolic geometry.

Straight ruled surfaces in the Heisenberg group

We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a straight ruled surface, i.e., it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.

Modulus of surface families and the radial stretch in the Heisenberg group

We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.

Quasiconformal mappings on the Heisenberg group: An overview

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension of the period defined using Chens iterated integrals and captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. As an application, we give an algorithm in proving nontriviality for a class of homologically trivial algebraic cycles obtained from special compact Riemann surfaces. The moduli space of compact Riemann surfaces is the space of all biholomorphism classes of compact Riemann surfaces. The harmonic volume can be regarded as an analytic section of a local system on the moduli space. It enables a quantitative study of the local structure of the moduli space. We explain basic concepts related to the harmonic volume and its applications of the moduli space.We give an overview of the proof for Mirzakhanis volume recursion for the Weil-Petersson volumes of the moduli spaces of genus

Complex hyperbolic Fenchel-Nielsen coordinates

g

The PU(2,1) configuration space of four points in S 3 and the cross-ratio variety

hyperbolic surfaces with

Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space.

n