References of "Bonsante, Francesco"
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See detailQuasicircles and width of Jordan curves in CP1
bonsante, francesco; danciger, jeffrey; maloni, sara et al

E-print/Working paper (2019)

We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in ... [more ▼]

We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles. [less ▲]

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See detailThe induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry
Bonsante, Francesco; Danciger, Jeff; Maloni, Sara et al

E-print/Working paper (2019)

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a ... [more ▼]

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature K∈[−1,0) and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes. [less ▲]

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See detailMinimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold
Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc UL

E-print/Working paper (2019)

Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree ... [more ▼]

Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in M. We prove the uniqueness of smooth minimizing maps from (S,h) to M in a given homotopy class. When (S,h) is fixed, smooth minimizing maps from (S,h) are described by a simple holomorphic data on S: a complex self-adjoint Codazzi tensor of determinant 1. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the holonomy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of F analoguous to the complex length. [less ▲]

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See detailOn the volume of anti-de Sitter maximal globally hyperbolic three-manifolds
Bonsante, Francesco; Seppi, Andrea; Tamburelli, Andrea UL

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 ▲]

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See detailA cyclic extension of the earthquake flow II
Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc UL

in Annales Scientifiques de l'Ecole Normale Supérieure (2015), 48(4), 811859

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See detailCollisions of particles in locally AdS spacetimes II. Moduli of globally hyperbolic spaces
Barbot, Thierry; Bonsante, Francesco; Schlenker, Jean-Marc UL

in Communications in Mathematical Physics (2014), 327(3), 691-735

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See detailA cyclic extension of the earthquake flow I
Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc UL

in Geometry & Topology (2013), 17(1), 157--234

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See detailFixed points of compositions of earthquakes
Bonsante, Francesco; Schlenker, Jean-Marc UL

in Duke Mathematical Journal (2012), 161(6), 1011--1054

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See detailCollisions of particles in locally AdS spacetimes I. Local description and global examples
Barbot, Thierry; Bonsante, Francesco; Schlenker, Jean-Marc UL

in Communications in Mathematical Physics (2011), 308(1), 147--200

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See detailMulti-black holes and earthquakes on Riemann surfaces with boundaries
Bonsante, Francesco; Krasnov, Kirill; Schlenker, Jean-Marc UL

in International Mathematics Research Notices (2011), (3), 487--552

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See detailMaximal surfaces and the universal Teichmüller space
Bonsante, Francesco; Schlenker, Jean-Marc UL

in Invent. Math. (2010), 182(2), 279--333

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See detailAdS manifolds with particles and earthquakes on singular surfaces
Bonsante, Francesco; Schlenker, Jean-Marc UL

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 ▲]

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See detailNotes on: ``Lorentz spacetimes of constant curvature'' [Geom. Dedicata 126 (2007), 3--45; MR2328921] by G. Mess
Andersson, Lars; Barbot, Thierry; Benedetti, Riccardo et al

in Geom. Dedicata (2007), 126

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