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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: 78 (3 UL)A quantum categorification of the Alexander polynomial Robert, Louis-Hadrien ; in Geometry and Topology (2021) Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page ... [more ▼] Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky. [less ▲] Detailed reference viewed: 29 (2 UL)Spherical CR uniformization of Dehn surgeries of the Whitehead link complement Acosta, Miguel in Geometry and Topology (2019), 23(5), 25932664 We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as a starting point the ... [more ▼] We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as a starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane H2C. We deform the Ford domain of Parker and Will in H2C in a one-parameter family. On one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer n≥4, and the spherical CR structure obtained for n=4 is the Deraux–Falbel spherical CR uniformization of the figure eight knot complement. [less ▲] Detailed reference viewed: 160 (3 UL)The convex core of quasifuchsian manifolds with particles ; Schlenker, Jean-Marc in Geometry and Topology (2014), 18(4), 2309--2373 Detailed reference viewed: 116 (4 UL)Dominating surface group representations and deforming closed AdS 3-manifolds Tholozan, Nicolas in Geometry and Topology (2014) Detailed reference viewed: 26 (1 UL)A cyclic extension of the earthquake flow I ; ; Schlenker, Jean-Marc in Geometry and Topology (2013), 17(1), 157--234 Detailed reference viewed: 120 (5 UL) |
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