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The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry ; ; et al in Geometry and Topology (in press) Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a ... [more ▼] Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature Kâˆˆ[âˆ’1,0) and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes. [less ▲] Detailed reference viewed: 66 (3 UL)Weakly Inscribed Polyhedra ; Schlenker, Jean-Marc in Transactions of the American Mathematical Society. Series B (in press) Detailed reference viewed: 50 (1 UL)Volumes of quasifuchsian manifolds Schlenker, Jean-Marc in Surveys in Differential Geometry, vol 24 (in press) Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between ... [more ▼] Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil-Petersson distance between the conformal metrics at infinity. [less ▲] Detailed reference viewed: 70 (8 UL)Bending laminations on convex hulls of anti-de Sitter quasicircles ; Schlenker, Jean-Marc in Proceedings of the London Mathematical Society (in press) Let λ− and λ+ be two bounded measured laminations on the hyperbolic disk H2, which "strongly fill" (definition below). We consider the left earthquakes along λ− and λ+, considered as maps from the ... [more ▼] Let λ− and λ+ be two bounded measured laminations on the hyperbolic disk H2, which "strongly fill" (definition below). We consider the left earthquakes along λ− and λ+, considered as maps from the universal Teichmüller space T to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism u:RP1→RP1, the boundary of the convex hull in AdS3 of its graph in RP1×RP1≃∂AdS3 is the disjoint union of two embedded copies of the hyperbolic plane, pleated along measured geodesic laminations. Our main result is that any pair of bounded measured laminations that "strongly fill" can be obtained in this manner. [less ▲] Detailed reference viewed: 73 (1 UL)Quasicircles and width of Jordan curves in CP1 ; ; et al in Bulletin of the London Mathematical Society (2021), 53(2), 507--523 We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in ... [more ▼] We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles. [less ▲] Detailed reference viewed: 72 (3 UL)On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces Schlenker, Jean-Marc E-print/Working paper (2021) The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K≥0 on the sphere is induced on the boundary of a unique convex body in $\R^3$. The ... [more ▼] The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K≥0 on the sphere is induced on the boundary of a unique convex body in $\R^3$. The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a ``dual'' statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later proved. We describe three conjectural generalizations of the Weyl problem in $\HH^3$ and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a number of recent results and well-known open problems can be considered as special cases. One focus is on convex domain having a ``thin'' asymptotic boundary, for instance a quasicircle -- this part of the problem is strongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a ``thick'' ideal boundary, for instance a disjoint union of disks -- here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete, convex disks of infinite area in $\HH^3$ and surfaces in hyperbolic ends -- with connections to questions on circle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recent works. [less ▲] Detailed reference viewed: 38 (0 UL)The Weyl problem for unbounded convex domains in $\HH^3$ Schlenker, Jean-Marc E-print/Working paper (2021) Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded ... [more ▼] Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the conformal structure on the boundary of $K$ in the Poincar\'e model of $\HH^3$, together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. We do not consider here the uniqueness of a convex subset with given boundary data. [less ▲] Detailed reference viewed: 22 (0 UL)A hyperbolic proof of Pascal's Theorem ; Schlenker, Jean-Marc in Mathematical Intelligencer (2021), 43(2), 130--133 We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Möbius, using hyperbolic geometry. Detailed reference viewed: 65 (0 UL)The prestige and status of research fields within mathematics Schlenker, Jean-Marc E-print/Working paper (2020) While the ``hierarchy of science'' has been widely analysed, there is no corresponding study of the status of subfields within a given scientific field. We use bibliometric data to show that subfields of ... [more ▼] While the ``hierarchy of science'' has been widely analysed, there is no corresponding study of the status of subfields within a given scientific field. We use bibliometric data to show that subfields of mathematics have a different ``standing'' within the mathematics community. Highly ranked departments tend to specialize in some subfields more than in others, and the same subfields are also over-represented in the most selective mathematics journals or among recipients of top prizes. Moreover this status of subfields evolves markedly over the period of observation (1984--2016), with some subfields gaining and others losing in standing. The status of subfields is related to different publishing habits, but some of those differences are opposite to those observed when considering the hierarchy of scientific fields. We examine possible explanations for the ``status'' of different subfields. Some natural explanations -- availability of funding, importance of applications -- do not appear to function, suggesting that factors internal to the discipline are at work. We propose a different type of explanation, based on a notion of ``focus'' of a subfield, that might or might not be specific to mathematics. [less ▲] Detailed reference viewed: 47 (1 UL)Polyhedra inscribed in a quadric ; ; Schlenker, Jean-Marc in Inventiones Mathematicae (2020), 221(1), 237-300 We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main ... [more ▼] We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the sphere and $\Gamma$ admits a Hamiltonian cycle. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout. [less ▲] Detailed reference viewed: 102 (7 UL)Flipping Geometric Triangulations on Hyperbolic Surfaces ; Schlenker, Jean-Marc ; in symposium on computational geometry (SoCG) (2020) We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with ... [more ▼] We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation. [less ▲] Detailed reference viewed: 23 (2 UL)Constant Gauss curvature foliations of AdS spacetimes with particles ; Schlenker, Jean-Marc in Transactions of the American Mathematical Society (2020), 373(6), 4013--4049 We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time N with particles (cone singularities of angles less than π along time-like curves), the ... [more ▼] We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time N with particles (cone singularities of angles less than π along time-like curves), the complement of the convex core in N admits a unique foliation by constant Gauss curvature surfaces. This extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe a parametrization of the space of convex GHM AdS metrics on a given manifold, with particles of given angles, by the product of two copies of the Teichm\"uller space of hyperbolic metrics with cone singularities of fixed angles. Finally, we use the results on K-surfaces to extend to hyperbolic surfaces with cone singularities of angles less than π a number of results concerning landslides, which are smoother analogs of earthquakes sharing some of their key properties. [less ▲] Detailed reference viewed: 61 (7 UL)Hyperideal polyhedra in the 3-dimensional anti-de Sitter space Chen, Qiyu ; Schlenker, Jean-Marc E-print/Working paper (2019) We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all ... [more ▼] We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all outside of AdS3 and whose edges all meet AdS3. We show that hyperideal polyhedra in AdS3 are uniquely determined by their combinatorics and dihedral angles, as well as by the induced metric on their boundary together with an additional combinatorial data, and describe the possible dihedral angles and the possible induced metrics on the boundary. [less ▲] Detailed reference viewed: 62 (2 UL)Hyperbolic ends with particles and grafting on singular surfaces ; Schlenker, Jean-Marc in Annales de L'Institut Henri Poincaré. Analyse Non Linéaire (2019), 36(1), 181-216 We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality ... [more ▼] We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichm\"uller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by "smooth grafting". [less ▲] Detailed reference viewed: 87 (11 UL)Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold ; ; Schlenker, Jean-Marc E-print/Working paper (2019) Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree ... [more ▼] Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in M. We prove the uniqueness of smooth minimizing maps from (S,h) to M in a given homotopy class. When (S,h) is fixed, smooth minimizing maps from (S,h) are described by a simple holomorphic data on S: a complex self-adjoint Codazzi tensor of determinant 1. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the holonomy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of F analoguous to the complex length. [less ▲] Detailed reference viewed: 36 (0 UL)Properness for circle packings and Delaunay circle patterns on complex projective structures Schlenker, Jean-Marc ; Yarmola, Andrew E-print/Working paper (2018) We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex ... [more ▼] We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper. [less ▲] Detailed reference viewed: 36 (4 UL)Delaunay Triangulations of Points on Circles ; ; Parlier, Hugo et al E-print/Working paper (2018) Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon ... [more ▼] Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm. [less ▲] Detailed reference viewed: 67 (10 UL)The renormalized volume and uniformisation of conformal structures ; ; Schlenker, Jean-Marc in Journal de l'institut de mathématiques de Jussieu (2018), 17(4), 853-912 Detailed reference viewed: 133 (16 UL)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: 37 (2 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: 43 (2 UL) |
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