Maximal surfaces and the universal Teichmüller space.
Francesco Bonsante,
Jean-Marc Schlenker.
To appear, Inventiones Mathematicae. 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.
To appear, Indiana Univ. Math. J.
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. 2010, doi: 10.1093/imrn/rnq070.
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.
To appear, Pacific J. Math.
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).
Higher Schlaefli Formulas and Applications II.
Vector-valued differential relations.
Jean-Marc Schlenker and Rabah Souam.
math.DG/0611499.
Int. Math. Res. Not. IMRN 2008, Art. ID rnn 068, 44 pp.
Text as pdf.The classical Schlaefli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds.
Circle patterns on singular surfaces.
Jean-Marc Schlenker.
math.DG/0601631.
Discr. Comput. Geom., 40(2008):1, 47-102.
Text as ps or
pdf.We consider ``hyperideal'' circle patterns, i.e. patterns of disks which do not cover the whole surface, which are associated to hyperideal hyperbolic polyhedra. The main result is that, on a Euclidean or hyperbolic surface with conical singularities, those hyperideal circle patterns are uniquely determined by the intersection angles of the circles and the singular curvatures. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in [Sch05], however the proof is completely different (and more intricate) since [Sch05] used a shortcut which is not available here.
On the renormalized volume of hyperbolic 3-manifolds.
Kirill Krasnov and Jean-Marc Schlenker.
math.DG/0607081.
Communications in Mathematical Physics 279:3 (2008), 637-668.
Text as pdf.The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate the geometrical meaning of this quantity. We propose a new regularization procedure based on surfaces equidistant to a given convex surface $\partial N$. The renormalized volume computed via this procedure is shown to be equal to what we call the $W$-volume of the convex region $N$ given by the usual volume of $N$ minus the quarter of the integral of the mean curvature over $\partial N$. The $W$-volume satisfies some remarkable properties. First, this quantity is self-dual in the sense explained in the paper. Second, it verifies some simple variational formulas analogous to the classical geometrical Schläfli identities. These variational formulas are invariant under a certain transformation that replaces the data at $\partial N$ by those at infinity of $M$. We use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of Takhtajan et al on the Kaehler potential on various moduli spaces.
Notes on a paper of Mess.
Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman,
François Labourie, Kevin P. Scannell, Jean-Marc Schlenker.
arXiv:0706.0640. Geometriae Dedicata 126:1 (2007), 47-70.
These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published.
Minimal surfaces and particles in 3-manifolds.
Kirill Krasnov and Jean-Marc Schlenker.
math.DG/0511441.
Geometriae dedicata, 126:1 (2007), 187-254.
Text as ps or
pdf.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 ``quasi-fuchsian'' 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 Teichm\"uller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichm\"uller 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 different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.
Small deformations of polygons and polyhedra.
Jean-Marc Schlenker.
math.DG/0410058.
Trans. Amer. Math. Soc.
359 (2007), 2155-2189.
Text as dvi, ps or
pdf.This paper describes some elementary properties of isometric first-order deformations of spherical or hyperbolic polygons. The first point is a very simple description of the first-order variations of the angles under such isometric first-order deformation. Then we define a quadratic invariant on isometric first-order deformations, and show that, for convex polygons, it has a striking positivity property. As a consequence, we find a new proof of the (classically known) infinitesimal rigidity of convex Euclidean polyhedra, and prove an infinitesimal rigidity statement for "Fuchsian" polyhedral surfaces in the Minkowski space.
Hyperbolic manifolds with convex boundary.
Jean-Marc Schlenker.
math.DG/0205305.
Inventiones math. 163(2006):1, 109-169.
Text as dvi, ps or
pdf. Abstract as dvi,
ps or pdf.
Hyperideal circle patterns.
Jean-Marc Schlenker.
math.GT/0407043.
Math.
Res. Lett., 12 (2005):1, 85-112.
Text as dvi, ps or
pdf. A short paper on some special circle patterns which are associated to hyperideal hyperbolic polyhedra. The main result is a description of the possible intersection angles of the circles in different situations. The proof follows from a result of Otal on the possible pleating lamination of hyperbolic convex cores.
A rigidity criterion for non-convex polyhedra.
Jean-Marc Schlenker.
math.DG/0301333.
Discrete
and Computational Geometry, 33 (2005):2, 207-221.
Text as dvi, ps or
pdf. Abstract as dvi,
ps or pdf.