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Constant Gaussian curvature foliations and Schläfli formulas of hyperbolic 3-manifolds Mazzoli, Filippo E-print/Working paper (2019) We study the geometry of the foliation by constant Gaussian curvature surfaces (S_k)_k of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary ... [more ▼] We study the geometry of the foliation by constant Gaussian curvature surfaces (S_k)_k of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizations of the space of hyperbolic ends, defined by Labourie in 1992 in terms of the geometry of the leaves S_k. We give a new description of the renormalized volume using the constant curvature foliation. We prove a generalization of McMullen's Kleinian reciprocity theorem, which replaces the role of the Schwarzian parametrization with Labourie's parametrizations. Finally, we describe the constant curvature foliation of a hyperbolic end as the integral curve of a time-dependent Hamiltonian vector field on the cotangent space to Teichmüller space, in analogy to the Moncrief flow for constant mean curvature foliations in Lorenzian space-times. [less ▲] Detailed reference viewed: 28 (2 UL)The dual volume of quasi-Fuchsian manifolds and the Weil-Petersson distance Mazzoli, Filippo E-print/Working paper (2019) Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold M is bounded by an explicit constant, depending only on the topology of M ... [more ▼] Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold M is bounded by an explicit constant, depending only on the topology of M, times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of M. [less ▲] Detailed reference viewed: 71 (1 UL)The dual Bonahon-Schläfli formula Mazzoli, Filippo E-print/Working paper (2018) Given a differentiable deformation of geometrically finite hyperbolic 3-manifolds (M_t)_t, the Bonahon-Schläfli formula expresses the derivative of the volume of the convex cores (CM_t)_t in terms of the ... [more ▼] Given a differentiable deformation of geometrically finite hyperbolic 3-manifolds (M_t)_t, the Bonahon-Schläfli formula expresses the derivative of the volume of the convex cores (CM_t)_t in terms of the variation of the geometry of its boundary, as the classical Schläfli formula does for the volume of hyperbolic polyhedra. Here we study the analogous problem for the dual volume, a notion that arises from the polarity relation between the hyperbolic space H^3 and the de Sitter space dS^3. The corresponding dual Bonahon-Schläfli formula has been originally deduced from Bonahon's work by Krasnov and Schlenker. Here, making use of the differential Schläfli formula and the properties of the dual volume, we give a (almost) self-contained proof of the dual Bonahon-Schläfli formula, without making use of Bonahon's original result. [less ▲] Detailed reference viewed: 64 (9 UL)Intertwining operators of the quantum Teichmüller space Mazzoli, Filippo E-print/Working paper (2016) Detailed reference viewed: 136 (21 UL) |
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