manifolds with boundary
Non-rigidity of spherical inversive distance circle packings.
Jiming Ma,
Jean-Marc Schlenker.
arXiv:1105.1469.
Discrete and Computational Geometry 47:3 (2012), 610--617.
We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.
Flippable tilings of constant curvature surfaces.
François Fillastre,
Jean-Marc Schlenker.
arXiv:1012.1594.
To appear, Illinois J. Math.
Text as pdf .We call ``flippable tilings'' of a constant curvature surface a tiling by ``black'' and ``white'' faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to ``flip'' the tiling by pushing all black faces forward on the left side and backward on the right side. Among those tilings we distinguish the ``symmetric'' ones, for which the metric on the surface does not change under the flip. We provide some existence statements, and explain how to parameterize the space of those tilings (with a fixed number of black faces) in different ways. For instance one can glue the white faces only, and obtain a metric with cone singularities which, in the hyperbolic and spherical case, uniquely determines a symmetric tiling. The proofs are based on the geometry of polyhedral surfaces in 3-dimensional spaces modeled either on the sphere or on the anti-de Sitter space.
Fixed points of compositions of earthquakes.
Francesco Bonsante, Jean-Marc Schlenker.
arXiv:0812.3471.
Duke Mathematical Journal 161(2012):6, 1011--1054.
Text as pdf, ps.Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.
Collisions of particles in locally AdS spacetimes I. Local description and global examples.
Thierry Barbot, Francesco Bonsante and Jean-Marc Schlenker.
arXiv:1010.3602.
Communications in Mathematical Physics 308(2011):1, 147-200.
Text as pdf.We investigate 3-dimensional globally hyperbolic AdS manifolds (or more generally constant curvature Lorentz 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 $\Gamma$). We then adapt to this setting some notions like global hyperbolicity which are natural for Lorentz manifolds, and construct some examples of globally hyperbolic AdS manifolds with interacting particles.
Combinatorial aspects of orthogonal group integrals.
Teodor Banica and Jean-Marc Schlenker.
arXiv:1011.2454. Intern. J. Math. 22:11 (2011), 1611-1646.
We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix of exponents $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the ``elementary expansion'' formula from the case $a\in M_{2\times q}(2\mathbb N)$ to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of a ``best algebraic normalization'' of $I(a)$, in the case $a\in M_{2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a\in M_{1\times 2}(\mathbb N)$, in relation with the Euler-Rodrigues formula. Most proofs use various combinatorial techniques.
Volume maximization and the extended hyperbolic space.
Feng Luo, Jean-Marc Schlenker.
arXiv:0908.2023.
Proc. Amer. Math. Soc. 140:3 (2012) 1053--1068.
Text as pdf, ps.We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space -- the natural extension of hyperbolic space by the de Sitter space -- except for the degenerate case where all simplices are Euclidean in a generalized sense.
Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as connected sum of manifolds admitting a complete hyperbolic metric, along embedded spheres (or projective planes) which are totally geodesic, space-like surfaces in the de Sitter part of the extended hyperbolic structure.
On polynomial integrals over the orthogonal group.
Teodor Banica,
Benoit Collins,
Jean-Marc Schlenker.
arXiv:0910.1258.
Journal of Combinatorial Theory A 118:3 (2011), 778-795.
Text as pdf .We consider integrals of type $\int_{O_n}u_{11}^{a_1}... u_{1n}^{a_n}u_{21}^{b_1}... u_{2n}^{b_n} du$, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials.
Maximal surfaces and the universal Teichmüller space.
Francesco Bonsante,
Jean-Marc Schlenker.
Inventiones Mathematicae 182(2010):279-333. arXiv:0911.4124.
Text as pdf .We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.
On orthogonal matrices maximizing the 1-norm.
Teodor Banica,
Benoit Collins,
Jean-Marc Schlenker.
arXiv:0901.2923.
Indiana Univ. Math. J. 59(2010):3, 839-856.
Text as pdf .For $U\in O(N)$ we have $||U||_1\leq N\sqrt{N}$, with equality if and only if $U=H/\sqrt{N}$, with $H$ Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1-norm on O(N). The main problem is to compute the $k$-th moment of the 1-norm, with $k\to\infty$, and we present a number of general comments in this direction.
Multi Black Holes and Earthquakes on Riemann surfaces with boundaries.
Francesco Bonsante, Kirill Krasnov, Jean-Marc Schlenker.
Int. Math. Res. Not. 2011 (2011):3, 487-552.
math.GT/0610429.
Abstract or pdf.We prove an ``Earthquake Theorem'' for hyperbolic metrics with geodesic boundary on a compact surfaces $S$ with boundary: the action of earthquakes on the enhanced Teichmüller space of $S$ is simply transitive. The proof rests on the geometry of ``multi-black holes'', which are 3-dimensional anti-de Sitter manifolds, topologically the product of a surface with boundary by an interval.
The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds.
Kirill Krasnov and Jean-Marc Schlenker.
arXiv:0907.2590.
To appear, Handbook of Teichmueller theory, vol III.
Text as pdf.We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichm\"uller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a K\"ahler potential for the Weil-Petersson metric on Teichm\"uller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends.
On the infinitesimal rigidity of polyhedra with vertices in convex position.
Ivan Izmestiev,
Jean-Marc Schlenker.
arXiv:0711.1981.
Pacific J. Math. 248(2010):1, 171-190.
Text as pdf.Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability.
The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite.
We study the shape of inflated surfaces introduced in~\cite{B1} and~\cite{P1}. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves which we describe explicitly as the solutions of a certain differential equation.
A symplectic map between hyperbolic and complex Teichmüller theory.
Kirill Krasnov, Jean-Marc Schlenker.
arXiv:0806.0010. Duke
Mathematical Journal 150(2009):2, 331-356.
Text as pdf.Let $S$ be a closed, orientable surface of genus at least $2$. The cotangent bundle of the ``hyperbolic'' Teichmüller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations, while the cotangent bundle of the ``complex'' Teichmüller space can be identified with $\CP$ through the Schwarzian derivative. We prove that the resulting map between the two cotangent spaces, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends.
Representations of quantum permutation algebras.
Teodor Banica,
Julien Bichon,
Jean-Marc Schlenker.
arXiv:0901.2331.
J. Funct. Anal. 257 (2009), 2864-2910.
Text as pdf .We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type $\pi:A_s(n)\to B(H)$. We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to $n=6$.
On weakly convex star-shaped polyhedra.
Jean-Marc Schlenker.
arXiv:0704.2901.
Text as pdf. Discrete
Mathematics 309(2009):20, 6139-6149..Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.
Quasi-Fuchsian manifolds with particles.
Sergiu Moroianu
and Jean-Marc Schlenker.
math.DG/0603441.
J. Differential Geom. 83:1 (2009), 75-129.
Text as ps or
pdf.We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $\pi$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure.
On the infinitesimal rigidity of weakly convex polyhedra.
Robert Connelly
and Jean-Marc Schlenker.
math.DG/0606681.
Text as pdf.
European Journal of Combinatorics, 31(2010):4, 1080-1090.
Special issue, Rigidity and related topics in Geometry.
The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by ``denting'' at most two edges at a common vertex, and suspensions with a natural subdivision.
AdS manifolds with particles and earthquakes on singular
surfaces.
Francesco Bonsante, Jean-Marc Schlenker.
math.GT/0609116.
Geom. Funct. Anal. 19:1 (2009), 41-82.
Text as pdf (two pictures missing), ps.We prove an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: any two points in the Teichmüller space are connected by a unique left earthquakes. This is strongly related to another result: the space of ``globally hyperbolic'' AdS manifolds with cone singularities (of given angle) along time-like geodesics is parametrized by the product of two copies of the Teichmüller space with some marked points (corresponding to the cone singularities).