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Quasicircles and width of Jordan curves in CP1 ; ; et al E-print/Working paper (2019) 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: 30 (0 UL)The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry ; ; et al E-print/Working paper (2019) 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: 42 (2 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: 27 (0 UL)On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds ; ; Tamburelli, Andrea in Geometric & Functional Analysis (2017) We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in ... [more ▼] We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in Teichmüller space of S provided by Mess’ parameterization - namely on two isotopy classes of hyperbolic metrics h and h' on S. The main result of the paper is that the volume coarsely behaves like the minima of the L1-energy of maps from (S, h) to (S, h'). The study of Lp-type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston’s Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behaviour is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil-Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from (S, h) to (S, h'), and to the minima of the holomorphic 1-energy. [less ▲] Detailed reference viewed: 112 (17 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: 136 (15 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: 83 (6 UL)A cyclic extension of the earthquake flow I ; ; Schlenker, Jean-Marc in Geometry & Topology (2013), 17(1), 157--234 Detailed reference viewed: 97 (4 UL)Fixed points of compositions of earthquakes ; Schlenker, Jean-Marc in Duke Mathematical Journal (2012), 161(6), 1011--1054 Detailed reference viewed: 83 (3 UL)Collisions of particles in locally AdS spacetimes I. Local description and global examples ; ; Schlenker, Jean-Marc in Communications in Mathematical Physics (2011), 308(1), 147--200 Detailed reference viewed: 67 (1 UL)Multi-black holes and earthquakes on Riemann surfaces with boundaries ; ; Schlenker, Jean-Marc in International Mathematics Research Notices (2011), (3), 487--552 Detailed reference viewed: 80 (2 UL)Maximal surfaces and the universal Teichmüller space ; Schlenker, Jean-Marc in Invent. Math. (2010), 182(2), 279--333 Detailed reference viewed: 126 (2 UL)AdS manifolds with particles and earthquakes on singular surfaces ; Schlenker, Jean-Marc in Geometric & Functional Analysis (2009), 19(1), 41--82 We prove an ``Earthquake theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: the action of the space of measured laminations on Teichm\"uller space ... [more ▼] We prove an ``Earthquake theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: the action of the space of measured laminations on Teichm\"uller space by left earthquakes is simply transitive. This is strongly related to another result: the space of ``globally hyperbolic'' AdS manifolds with cone singularities along time-like geodesics is parametrized by the product two copies of Teichm\"uller space with some marked points (corresponding to the cone singularities). [less ▲] Detailed reference viewed: 88 (4 UL)Notes on: ``Lorentz spacetimes of constant curvature'' [Geom. Dedicata 126 (2007), 3--45; MR2328921] by G. Mess ; ; et al in Geom. Dedicata (2007), 126 Detailed reference viewed: 141 (0 UL) |
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