Complex hyperbolic quasi-Fuchsian groups

Cosh-Gordon equation and quasi-Fuchsian groups

We show propositions in favour of relations between the phase space of 3D gravity, moduli of quasi-Fuchsian groups, global solutions of cosh-Gordon equations and minimal surfaces in hyperbolic spaces.

Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography

We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasiFuchsian deformation space, is an antiderivative of a 1-form ...

On Hyperbolic Polyhedra Arising as Convex Cores of Quasi-Fuchsian Punctured Torus Groups

Minimal Lagrangian Surfaces in Ch and Representations of Surface Groups into Su(2, 1)

We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface in CH from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero sect...

Complex Hyperbolic Fenchel-nielsen Coordinates

Let Σ be a closed, orientable surface of genus g. It is known that the SU(2, 1) representation variety of π1(Σ) has 2g− 3 components of (real) dimension 16g− 16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety th...