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Constant curvature surfaces and volumes of convex co-compact hyperbolic manifolds Mazzoli, Filippo Doctoral thesis (2020) We investigate the properties of various notions of volume for convex co-compact hyperbolic 3-manifolds, and their relations with the geometry of the Teichmüller space. We prove a first-order variation ... [more ▼] We investigate the properties of various notions of volume for convex co-compact hyperbolic 3-manifolds, and their relations with the geometry of the Teichmüller space. We prove a first-order variation formula for the dual volume of the convex core, as a function over the space of quasi-isometric deformations of a convex co-compact hyperbolic 3-manifold. For quasi-Fuchsian manifolds, we show that the dual volume of the convex core is bounded from above by a linear function of the Weil-Petersson distance between the pair of hyperbolic structures on the boundary of the convex core. We prove that, as we vary the convex co-compact structure on a fixed hyperbolic 3-manifold with incompressible boundary, the infimum of the dual volume of the convex core coincides with the infimum of the Riemannian volume of the convex core. We study various properties of the foliation by constant Gaussian curvature surfaces (k-surfaces) of convex co-compact hyperbolic 3-manifolds. We present a description of the renormalized volume of a quasi-Fuchsian manifold in terms of its foliation by k-surfaces. We show the existence of a Hamiltonian flow over the cotangent space of Teichmüller space, whose flow lines corresponds to the immersion data of the k-surfaces sitting inside a fixed hyperbolic end, and we determine a generalization of McMullen’s Kleinian reciprocity, again by means of the constant Gaussian curvature surfaces foliation. [less ▲] Detailed reference viewed: 164 (12 UL)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: 61 (5 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: 92 (6 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: 81 (10 UL)Intertwining operators of the quantum Teichmüller space Mazzoli, Filippo E-print/Working paper (2016) Detailed reference viewed: 149 (24 UL) |
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