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Anti-de Sitter geometry: convex domains, foliations and volume Tamburelli, Andrea Doctoral thesis (2018) We study various aspects of the geometry of globally hyperbolic anti-de Sitter 3-manifolds. For manifolds with convex space-like boundaries, homeomorphic to the product of a closed, connected and oriented ... [more ▼] We study various aspects of the geometry of globally hyperbolic anti-de Sitter 3-manifolds. For manifolds with convex space-like boundaries, homeomorphic to the product of a closed, connected and oriented surface of genus at least two with an interval, we prove that every pair of metrics with curvature less than -1 on the surface can be realised on the two boundary components. For globally hyperbolic maximal compact (GHMC) anti-de Sitter manifolds, we study various geometric quantities, such as the volume, the Hausdorff dimension of the limit set, the width of the convex core and the Holder exponent of the manifold, in terms of the parameters that describe the deformation space of GHMC anti-de Sitter structures. Moreover, we prove existence and uniqueness of a foliation by constant mean curvature surfaces of the domain of dependence of any quasi-circle in the boundary at infinity of anti-de Sitter space. [less ▲] Detailed reference viewed: 39 (5 UL)Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe Tamburelli, Andrea E-print/Working paper (2017) We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an ... [more ▼] We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane. [less ▲] Detailed reference viewed: 51 (2 UL)Entropy degeneration of globally hyperbolic maximal compact anti-de Sitter structures Tamburelli, Andrea E-print/Working paper (2017) Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on S×ℝ by the cotangent bundle of the Teichmüller space of S, we study how some geometric quantities, such as the ... [more ▼] Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on S×ℝ by the cotangent bundle of the Teichmüller space of S, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponent, degenerate along rays of quadratic differentials. [less ▲] Detailed reference viewed: 37 (1 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: 59 (14 UL)Constant mean curvature foliation of globally hyperbolic (2+1)-spacetime with particles ; Tamburelli, Andrea E-print/Working paper (2017) Let M be a globally hyperbolic maximal compact 3-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that M admits a unique foliation by constant mean ... [more ▼] Let M be a globally hyperbolic maximal compact 3-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that M admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularities of angles less than π along time-like geodesics). [less ▲] Detailed reference viewed: 44 (0 UL)Constant mean curvature foliation of domain of dependence in AdS3 Tamburelli, Andrea in Transactions of the American Mathematical Society (2016) We prove that, given an acausal curve in the boundary at infinity of Anti-de Sitter space which is the graph of a quasi-symmetric homeomorphism, there exists a foliation of its domain of dependence by ... [more ▼] We prove that, given an acausal curve in the boundary at infinity of Anti-de Sitter space which is the graph of a quasi-symmetric homeomorphism, there exists a foliation of its domain of dependence by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of the quasi-symmetric homeomorphism we started with. [less ▲] Detailed reference viewed: 82 (11 UL)Prescribing metrics on the boundary of AdS 3-manifolds Tamburelli, Andrea in International Mathematics Research Notices (2016) We prove that given two metrics g+ and g− with curvature κ<−1 on a closed, oriented surface S of genus τ≥2, there exists an AdS manifold N with smooth, space-like, strictly convex boundary such that the ... [more ▼] We prove that given two metrics g+ and g− with curvature κ<−1 on a closed, oriented surface S of genus τ≥2, there exists an AdS manifold N with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of ∂N are equal to g+ and g−. Using the duality between convex space-like surfaces in AdS3, we obtain an equivalent result about the prescription of the third fundamental form. [less ▲] Detailed reference viewed: 100 (13 UL) |
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