manifolds with boundary
circle patterns, polyhedral geometry,
discrete geometry
Hadamard matrices, integration over O(n), etc
Collisions of particles in locally AdS spacetimes II. Moduli of globally hyperbolic spaces
Thierry Barbot, Francesco Bonsante and Jean-Marc Schlenker.
arXiv:1202.5753.
Text as pdf.We investigate 3-dimensional globally hyperbolic AdS manifolds containing ``particles'', i.e., cone singularities of angles less than $2\pi$ along a time-like graph $\Gamma$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.
The renormalized volume and the volume of the convex core of quasifuchsian manifolds.
Jean-Marc Schlenker.
arXiv:1109.6663.
We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we show that holomorphic disks in Teichm\"uller space which are large enough must have ``enough'' negative curvature.
A cyclic extension of the earthquake flow.
Francesco Bonsante,
Gabriele Mondello,
Jean-Marc Schlenker.
arXiv:1106.0525.
Let $\cT$ be Teichm\"uller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston's Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on $\cT\times \cT$ extends to the product of two copies of the universal Teichm\"uller space.
What does it take to become a good mathematician?
Pierre Dubois,
Jean-Charles Rochet,
Jean-Marc Schlenker.
Working paper
TSE 10-160
and
IDEI 606
,
may 2010.
Using an exhaustive database on academic publications in mathematics, we study the patterns of productivity by world mathematicians over the period 1984-2006. We uncover some surprising facts, such as the absence of age related decline in productivity and the relative symmetry of international movements, rejecting the presumption of a massive "brain drain" towards the U.S. Looking at the U.S. academic market in mathematics, we analyze the determinants of success by top departments. In conformity with recent studies in other fields, we find that selection effects are much stronger than local interaction effects: the best departments are most successful in hiring the most promising mathematicians, but not necessarily at stimulating positive externalities among them. Finally we analyze the impact of career choices by mathematicians: mobility almost always pays, but early specialization does not.
The convex core of quasifuchsian manifolds with particles.
Cyril Lecuire, Jean-Marc Schlenker.
arXiv:0909.4182.
Text as pdf, ps.We consider quasifuchsian manifolds with ``particles'', i.e., cone singularities of fixed angle less than $\pi$ going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism from the space of quasifuchsian metrics with $n$ particles (of fixed angle) and the product of two copies of the Teichm\"uller space of a surface with $n$ marked points. This is analoguous to the Bers theorem in the non-singular case. Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in [BO] in the non-singular case.
Collisions of particles in locally AdS spacetimes.
Thierry Barbot, Francesco Bonsante and Jean-Marc Schlenker.
arXiv:0905.1823.
Text as pdf.We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph $\Gamma$. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than $2\pi$ on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of $\Gamma$). The local geometry near an "interaction point" (a vertex of the singular locus) has a simple geometric description in terms of polyhedra in the extension of hyperbolic 3-space by the de Sitter space.
We then concentrate on spaces containing only (interacting) massive particles. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.