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Conformal actions of higher-rank lattices on pseudo-Riemannian manifolds Pecastaing, Vincent in Geometric and Functional Analysis (2020), 30 We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which ... [more ▼] We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group ([33]). When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of [1] and [17] in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exceptional groups. Our work is strongly inspired by the recent breakthrough of Brown, Fisher and Hurtado on Zimmer’s conjecture [7]. [less ▲] Detailed reference viewed: 74 (4 UL)Flows on the PSL(V)-Hitchin component Sun, Zhe ; ; in Geometric and Functional Analysis (2020), 30 In this article we define new flows on the Hitchin components for PSL(n,R). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other ... [more ▼] In this article we define new flows on the Hitchin components for PSL(n,R). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component. [less ▲] Detailed reference viewed: 51 (5 UL)On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds ; ; Tamburelli, Andrea in Geometric and 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: 137 (18 UL)Non-universality of nodal length distribution for arithmetic random waves ; Peccati, Giovanni ; Rossi, Maurizia et al in Geometric and Functional Analysis (2016), 26(3), 926-960 Detailed reference viewed: 150 (21 UL)Stein's method, logarithmic Sobolev and transport inequalities ; Nourdin, Ivan ; Peccati, Giovanni in Geometric and Functional Analysis (2015), 25 Detailed reference viewed: 264 (17 UL)AdS manifolds with particles and earthquakes on singular surfaces ; Schlenker, Jean-Marc in Geometric and 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: 118 (5 UL) |
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