The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

Given a smooth spacelike surface $\Sigma$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $\rho:\pi_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $\Sigma$ a diffeomorphism $\phi_\Sigma$ of $S$. It turns out that $\phi_\Sigma$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $\rho$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that $\phi_\Sigma$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$.Comment: 20 page

On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Given a quasisymmetric homeomorphism $\varphi$ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension $f_\varphi:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies $\log K(f_\varphi)\leq C||\varphi||$, where $||\varphi||$ denotes the cross-ratio norm. We give constraints on the value of an optimal such constant $C$, and discuss possible lower inequalities, by studying two one-parameter families of minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio norm.Comment: 25 pages. Results of Theorem A improved. Several mistakes corrected, Remark 4.9 added, general exposition improve

Fibered spherical 3-orbifolds

In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of $\mathrm{SO}(4)$, this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we deal with the fibered case and in particular we give explicit formulae relating the finite subgroups of $\mathrm{SO}(4)$ with the invariants of the corresponding fibered 3-orbifolds. This allows to deduce directly from the algebraic classification topological properties of spherical 3-orbifolds.Comment: 27 pages, 6 figures. Several misprint corrected, improved expositio

On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry

Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichm\"uller space of $S$, if $S$ is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relation between the Mess parameters and the embedding data of any Cauchy surface. This relation is pointed out by using some specific properties of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new Lorentzian proof of Goldman's celebrated result about the coincidence of the Weil-Petersson symplectic form and the Goldman pairing. In the second part of the paper we use this machinery to get a classification of globally hyperbolic flat space-times with particles of angles in $(0,2\pi)$ containing a uniformly convex Cauchy surface. The analogue of Mess' result is achieved showing that the corresponding moduli space is the tangent bundle of the Teichm\"uller space of a punctured surface. To generalize the theory in the case of particles, we deepen the study of Codazzi tensors on hyperbolic surfaces with cone singularities, proving that the well-known decomposition of a Codazzi tensor in a harmonic part and a trivial part can be generalized in the context of hyperbolic metrics with cone singularities.Comment: 49 pages, 4 figure

Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichm\"uller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichm\"uller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.Comment: 21 pages, 11 figure

Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four

We provide the first examples of geometric transition from hyperbolic to Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboctahedron. We show the existence of a similar family of collapsing Anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope.Comment: 50 pages, 27 figures. Part 3 of the previous version has been removed and will be part of a new preprint to appear soo

Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space

We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $\Gamma$ is the graph of a quasisymmetric homeomorphism of the circle if and only if the $K$-surfaces have bounded principal curvatures. Moreover in this case a uniqueness result holds. The proofs rely on a well-known correspondence between spacelike surfaces in Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane. In fact, an important ingredient is a representation formula, which reconstructs a spacelike surface from the associated area-preserving diffeomorphism. Using this correspondence we then deduce that, for any fixed $\theta\in(0,\pi)$, every quasisymmetric homeomorphism of the circle admits a unique extension which is a $\theta$-landslide of the hyperbolic plane. These extensions are quasiconformal.Comment: 47 pages, 18 figures. More details added to Remark 4.14, Remark 6.2 and Theorem 7.8 Step 2. Several references added and typos corrected. Final version. To appear in Journal of Topolog