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https://arxiv.org/abs/2101.07083 El Emam, Christian ; E-print/Working paper (2021) We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an ... [more ▼] We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions. [less ▲] Detailed reference viewed: 80 (1 UL)On the Gauss map of equivariant immersions in hyperbolic space El Emam, Christian ; E-print/Working paper (2020) Given an oriented immersed hypersurface in hyperbolic space H^{n+1}, its Gauss map is defined with values in the space of oriented geodesics of H^{n+1}, which is endowed with a natural para-Kähler ... [more ▼] Given an oriented immersed hypersurface in hyperbolic space H^{n+1}, its Gauss map is defined with values in the space of oriented geodesics of H^{n+1}, which is endowed with a natural para-Kähler structure. In this paper we address the question of whether an immersion G of the universal cover of an n-manifold M, equivariant for some group representation of π1(M) in Isom(H^{n+1}), is the Gauss map of an equivariant immersion in H^{n+1}. We fully answer this question for immersions with principal curvatures in (−1,1): while the only local obstructions are the conditions that G is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for M compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms [less ▲] Detailed reference viewed: 30 (0 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) |
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