Space Of Projective Structures Research Articles

On the symplectic structure over a moduli space of orbifold projective structures

Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmuller space for S. The total space of the holomorphic cotangent bundle of the Teichmuller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification is proved to be compatible with these symplectic structures. Similar results are obtained for biholomorphisms constructed using uniformizations provided by Schottky groups and Earle's version of simultaneous uniformization.

Exotic projective structures and quasi-Fuchsian space, II

Let $P(S)$ be the space of projective structures on a closed surface $S$ of genus $g >1$ and let $Q(S)$ be the subset of $P(S)$ of projective structures with quasifuchsian holonomy. It is known that $Q(S)$ consists of infinitely many connected components. In this paper, we will show that the closure of any exotic component of $Q(S)$ is not a topological manifold with boundary and that any two components of $Q(S)$ have intersecting closures.

Circle Packings on Surfaces with Projective Structures

The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σg of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1,0 or −1 on the surface depending on whether g = 0,1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing contains a neigborhood of the Koebe-Andreev-Thurston structure homeomorphic to ℝ6g−6. We furthemore show that if a circle packing consists of one circle, then the space is globally homeomorphic to ℝ6g−6 and that the circle packing is rigid.

Higher dimensional uniformisation and W-geometry

We formulate the uniformisation problem underlying the geometry of W n -gravity using the differential equation approach to W-algebras. We construct W n -space (analogous to superspace in supersymmetry) as an ( n − 1)-dimensional complex manifold using isomonodromic deformations of linear differential equations. The W n -manifold is obtained by the quotient of a Fuchsian subgroup of PSL( n, R ) which acts properly discontinuously on a simply connected domain in C P n − 1. The requirement that a deformation be isomonodromic furnishes relations whic enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W n -manifold. We discuss how the Teichmüller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W n -manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all “generic” W-algebras.