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

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

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $\eta$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $\mu$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $\mu$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $\Gamma$ of isometries of $\mathbb{H}^d$, given a $\Gamma$ invariant Radon measure $\mu$, given a isometry group $\Gamma_{\tau}$ of Minkowski space, with $\Gamma$ as linear part, there exists a unique convex set with area measure $\mu$, invariant under the action of $\Gamma_{\tau}$. The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Amp\`ere equation. The existence part can be translated as an existence result for Monge--Amp\`ere equation. The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $\Gamma_{\tau}=\Gamma$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $\Gamma_\tau$-invariant surface of constant Gauss-Kronecker curvature equal to $1$

Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Amp\`ere equation $\det D^2 u(z)=(1/\psi(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=\varphi$, for $\psi$ a smooth positive function and $\varphi$ a bounded lower semicontinuous function. We then prove that a domain of dependence $D$ contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function $\varphi$ is in the Zygmund class. Moreover in this case the surface of constant curvature $K$ contained in $D$ has bounded principal curvatures, for every $K<0$. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of $\partial \mathbb{D}$. Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature $K$, as $K$ varies in $(-\infty,0)$.Comment: 45 pages, 17 figures. Final version, improved presentation and details of some proof

AdS manifolds with particles and earthquakes on singular surfaces

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``particles'' -- cone singularities (of given angle) along time-like lines -- is parametrized by the product of two copies of the Teichm\"uller space with some marked points (corresponding to the cone singularities). The two statements are proved together.Comment: 18 pages, several figures. v2: improved exposition, several correction