Doctoral thesis (Dissertations and theses)
Constant curvature surfaces and volumes of convex co-compact hyperbolic manifolds
Mazzoli, Filippo
2020
 

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Keywords :
Teichmüller theory; Kleinian groups; hyperbolic manifolds; convex core; dual volume; renormalized volume; hyperbolic geometry; quasi-Fuchsian groups; Bonahon-Schlafli formula; dual Bonahon-Schlafli formula
Abstract :
[en] 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.
Research center :
Department of Mathematics
Disciplines :
Mathematics
Author, co-author :
Mazzoli, Filippo ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Language :
English
Title :
Constant curvature surfaces and volumes of convex co-compact hyperbolic manifolds
Defense date :
23 July 2020
Number of pages :
xiv, 128
Institution :
Unilu - University of Luxembourg, Esch-sur-Alzette, Luxembourg
Degree :
Docteur en Mathématiques
FnR Project :
FNR10949314 - Geometric And Stochastic Methods In Mathematics And Applications, 2015 (01/10/2016-31/03/2023) - Gabor Wiese
Funders :
University of Luxembourg - UL
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since 27 July 2020

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