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

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 $1$

Multi Black Holes and Earthquakes on Riemann surfaces with boundaries

We prove an "Earthquake Theorem" for hyperbolic metrics with geodesic boundary on a compact surfaces $S$ with boundary: given two hyperbolic metrics with geodesic boundary on a surface with $k$ boundary components, there are $2^k$ right earthquakes transforming the first in the second. An alternative formulation arises by introducing the enhanced Teichmueller space of S: We prove that any two points of the latter are related by a unique right earthquake. The proof rests on the geometry of ``multi-black holes'', which are 3-dimensional anti-de Sitter manifolds, topologically the product of a surface with boundary by an interval.Comment: 29 pages, several figures. v2: corrections, more detailed arguments, etc. Update to v3 was an error while trying to update another preprint, v3 is not the right file. v4 reverts to v2. v5: streamlined introduction, various improvments in the expositio

Fixed points of compositions of earthquakes

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichm\"uller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichm\"uller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.Comment: 19 pages, 1 figure. v2: 21 pages, 3 figures. v2 is a substantial rewrite, with simpler proofs and better explanations, some corrections. v3: further improvements in the expositio