Related papers: Regular globally hyperbolic maximal anti-de Sitter…
We prove that the Teichm\"uller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that that the…
In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like…
We present two different quantum deformations for the (anti)de Sitter algebras and groups. The former is a non-standard (triangular) deformation of SO(4,2) realized as the conformal group of the (3+1)D Minkowskian spacetime, while the…
We determine optimal inequalities for the systole of all hyperbolic compact surfaces of caracteristic -1. First, we study the geometry and topology of these surfaces. Then, we describe the action of modular groups on Teichm\"{u}ller spaces.…
We provide a partial characterization of the conformal infinity of asymptotically de Sitter spacetimes by deriving constraints that relate the asymptotics of the stress-energy tensor with conformal geometric data. The latter is captured…
We give a review of our joint work on strict deformation of BHTZ 2+1 black holes \cite{BRS02,BDHRS03}. However some results presented here are not published elsewhere, and an effort is made for enlightening the instrinsical aspect of the…
Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…
In the thesis, results presented in various papers involving the author are reviewed. First, general results about consistent deformations at first order (i.e. cubic) using the antifield formalism in Minkowski spacetime are gathered.…
In this note we prove infinite dimensionality of the Teichm\"uller space of a hyperbolic Riemann surface lamination of a compact space having a simply connected leaf.
In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and…
Based on Teichm\"uller theory, we construct a degenerating family $\overline{Y}_g^{orb} \rightarrow \overline{M}_g^{orb}$ over the Deligne-Mumford compactification of the moduli space with the natural orbifold structure such that any…
We study an $\mathcal{N}=2$, pure $U(1)$ SYM theory on a Gibbons-Hawking space $\Omega$-deformed using the $U(1)$ isometry. The resultant 3D theory, after an appropriate "Nekrasov-Witten" change of variables, is asymptotically equivalent to…
In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface $\overline{X}$ with prescribed singularities. This…
Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic…
The authors study singular points of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 and the geometry of hypersurfaces and use them for construction of an invariant normalization and an invariant affine connection of lightlike…
We consider contracting flows in $(n+1)$-dimensional hyperbolic space and expanding flows in $(n+1)$-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding…
Given a compact orientable surface with finitely many punctures $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential unoriented simple closed curves in $\Sigma$. We determine a complete set of relations for a function…
Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…
Let $\lambda_-$ and $\lambda_+$ be two bounded measured laminations on the hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below). We consider the left earthquakes along $\lambda_-$ and $\lambda_+$, considered as maps from…
These notes describe representations of the universal cover of $\mathrm{SL}(2,\mathbb{R})$ with a view toward applications in physics. Spinors on the hyperbolic plane and the two-dimensional anti-de Sitter space are also discussed.