Jean-Marc Schlenker

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.

Compactly supported bidimensional wavelet bases with hexagonal symmetry

AbstractWe study a class of subband coding schemes allowing perfect reconstruction for a bidimensional signal sampled on the hexagonal grid. From these schemes we construct biorthogonal wavelet bases ofL2(R2) which are compactly supported and such that the sets of generating functionsψ1,ψ2,ψ3 for the synthesis and

Hyperbolic manifolds with convex boundary

\tilde \psi _1 , \tilde \psi _2 , \tilde \psi _3 ,

Fixed points of compositions of earthquakes

for the analysis, as well as the scaling functions φ and

A Rigidity Criterion for Non-Convex Polyhedra

\tilde \varphi

Circle Patterns on Singular Surfaces

, are globally invariant by a rotation of 2π/3. We focus on the particular case of linear splines and we discuss how to obtain a higher regularity. We finally present the possibilities of sharp angular frequency resolution provided by these new bases.