manifolds with boundary
circle patterns, polyhedral geometry,
discrete geometry
Recovering the geometry of a flat spacetime from a background radiation.
Francesco Bonsante,
Catherine Meusburger,
Jean-Marc Schlenker.
arXiv:1302.6966.
We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions, where a uniform light signal is emitted on the $r$-level surface of the cosmological time for $r\to 0$. We show that the intensity of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a purely classical model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2+1, we show that this observed intensity function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this intensity function in a few simple examples.
Analytic aspects of the circulant Hadamard conjecture.
Teodor Banica,
Ion Nechita,
Jean-Marc Schlenker.
arXiv:1212.3589.
We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here these questions, with some results and conjectures.
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the conformal class at infinity determined by $g$, we denote it by ${\rm Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional admitting a "Polyakov type" formula in the conformal class $[h_0]$ and we describe the critical points as solutions of some non-linear equation $v_n(h_0)={\rm const}$, satisfied in particular by Einstein metrics. In dimension $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while in dimension $n=4$ this amounts to solving the $\sigma_2$-Yamabe problem. Next, we consider the variation of ${\rm Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric $h_0^t$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\pl M}v_n(h){\rm dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to Identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space $\mc{T}(\pl M)$ of conformal structures on $\pl M$. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.
A cyclic extension of the earthquake flow II.
Francesco Bonsante,
Gabriele Mondello,
Jean-Marc Schlenker.
arXiv:1208.1738.
The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichm\"uller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichm\"uller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.
Some questions on anti-de Sitter geometry.
Thierry Barbot,
Francesco Bonsante,
Jeff Danciger, William M. Goldman, François Guéritaud,
Fanny Kassel, Kirill Krasnov, Jean-Marc Schlenker, Abdelghani Zeghib.
arXiv:1205.6103.
Text as pdf.We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature $-1$. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.
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.
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.