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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: 112 (17 UL)Non-universality of nodal length distribution for arithmetic random waves ; Peccati, Giovanni ; Rossi, Maurizia et al in Geometric & Functional Analysis (2016), 26(3), 926-960 Detailed reference viewed: 117 (18 UL)Stein's method, logarithmic Sobolev and transport inequalities ; Nourdin, Ivan ; Peccati, Giovanni in Geometric & Functional Analysis (2015), 25 Detailed reference viewed: 203 (16 UL)AdS manifolds with particles and earthquakes on singular surfaces ; Schlenker, Jean-Marc in Geometric & Functional Analysis (2009), 19(1), 41--82 We prove an ``Earthquake theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: the action of the space of measured laminations on Teichm\"uller space ... [more ▼] We prove an ``Earthquake theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: the action of the space of measured laminations on Teichm\"uller space by left earthquakes is simply transitive. This is strongly related to another result: the space of ``globally hyperbolic'' AdS manifolds with cone singularities along time-like geodesics is parametrized by the product two copies of Teichm\"uller space with some marked points (corresponding to the cone singularities). [less ▲] Detailed reference viewed: 88 (4 UL) |
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