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Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds. Schlenker, Jean-Marc E-print/Working paper (2017) The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the ... [more ▼] The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. \par We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincar\'e-Einstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a "constant curvature" space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and Monge-Amp\`ere equations. Notes aiming at clarifying the relations between different points of view and introducing one new notion, no real result. Not intended to be submitted at this point [less ▲] Detailed reference viewed: 50 (3 UL)Higher signature Delaunay decompositions ; ; Schlenker, Jean-Marc E-print/Working paper (2016) A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay ... [more ▼] A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles. [less ▲] Detailed reference viewed: 56 (2 UL)Polyhedra inscribed in a quadric and anti-de Sitter geometry Schlenker, Jean-Marc in Oberwolfach Reports (2016) A short survey on recent results concerning polyhedra inscribed in quadrics. Detailed reference viewed: 81 (4 UL)Small circulant complex Hadamard matrices of Butson type ; Schlenker, Jean-Marc in European Journal of Combinatorics (2016), 51 We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for ... [more ▼] We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pqn=p+q,l=pq with p,qp,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,ln,l. [less ▲] Detailed reference viewed: 156 (10 UL)Variétés lorentziennes plates vues comme limites de variétés anti-de Sitter, d'après Danciger, Guéritaud et Kassel Schlenker, Jean-Marc in Astérisque (2016), 380 A survey on the recent work of Danciger, Gu\'eritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by ... [more ▼] A survey on the recent work of Danciger, Gu\'eritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by (non-abelian) free groups acting propertly discontinuously. Goldman, Labourie and Margulis have shown that they are determined by a convex co-compact hyperbolic surface S along with a first-order deformation of the metric which uniformly decreases the lengths of closed geodesics. Danciger, Gu\'eritaud and Kassel show that those space-times are principal ℝ-bundles over S with time-like geodesics as fibers, that they are homeomorphic to the interior of a handlebody, and that they admit a fundamental domain bounded by crooked planes. To obtain those results they show that those Margulis space-times are "infinitesimal" versions of 3-dimensional anti-de Sitter manifolds, and are lead to introduce a new parameterization of the space of deformations of a hyperbolic surface that increase the lengths of all closed geodesics. [less ▲] Detailed reference viewed: 91 (3 UL)Anti-de Sitter space: from physics to geometry Schlenker, Jean-Marc in CMS Notes (2015), 47(3), 14-15 A survey of recent developments in anti-de Sitter geometry. Detailed reference viewed: 194 (8 UL)Polyhedra inscribed in a hyperboloid and anti-de Sitter geometry Schlenker, Jean-Marc Speeches/Talks (2015) Detailed reference viewed: 51 (1 UL)Three applications of anti-de Sitter geometry Schlenker, Jean-Marc Speeches/Talks (2015) Detailed reference viewed: 61 (5 UL)The renormalized volume of quasifuchsian manifolds Schlenker, Jean-Marc Speeches/Talks (2015) Detailed reference viewed: 60 (4 UL)Polyèdres inscrits dans des quadriques Schlenker, Jean-Marc Speeches/Talks (2015) Detailed reference viewed: 42 (6 UL)A cyclic extension of the earthquake flow II ; ; Schlenker, Jean-Marc in Annales Scientifiques de l'Ecole Normale Supérieure (2015), 48(4), 811859 Detailed reference viewed: 170 (16 UL)Productivity and Mobility in Academic Research: Evidence from Mathematicians ; ; Schlenker, Jean-Marc in Scientometrics (2014), 98(3), 1669-1701 Detailed reference viewed: 141 (3 UL)Submatrices of Hadamard matrices: complementation results ; ; Schlenker, Jean-Marc in Electronic Journal of Linear Algebra (2014), 27 Detailed reference viewed: 119 (9 UL)Recovering the geometry of a flat spacetime from a background radiation ; ; Schlenker, Jean-Marc in Annales Henri Poincare (2014), 15(9), 1733-1799 Detailed reference viewed: 125 (4 UL)Collisions of particles in locally AdS spacetimes II. Moduli of globally hyperbolic spaces ; ; Schlenker, Jean-Marc in Communications in Mathematical Physics (2014), 327(3), 691-735 Detailed reference viewed: 119 (6 UL)Analytic aspects of the circulant Hadamard conjecture ; ; Schlenker, Jean-Marc in Annales mathématiques Blaise Pascal (2014), 21 Detailed reference viewed: 134 (8 UL)The convex core of quasifuchsian manifolds with particles ; Schlenker, Jean-Marc in Geometry and Topology (2014), 18(4), 2309--2373 Detailed reference viewed: 129 (5 UL)Symplectic Wick rotations between moduli spaces of 3-manifolds ; Schlenker, Jean-Marc in Annali della Scuola Normale Superiore di Pisa: Classe di Scienze (2014) Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S ... [more ▼] Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called ``Wick rotations'', defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic. [less ▲] Detailed reference viewed: 76 (4 UL)The renormalized volume and the volume of the convex core of quasifuchsian manifolds Schlenker, Jean-Marc in Mathematical Research Letters (2013), 20(4), 773-786 Detailed reference viewed: 133 (4 UL)A cyclic extension of the earthquake flow I ; ; Schlenker, Jean-Marc in Geometry and Topology (2013), 17(1), 157--234 Detailed reference viewed: 146 (5 UL) |
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