arXiv subjects
Publications and source records attributed to Francesco Bonsante.
We prove that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.
We prove four results towards a description, in terms of the null support function, of the set of isometric embeddings of the hyperbolic plane into Minkowski 3-space. We show that for sufficiently tame null support function, the corresponding entire surface of constant curvature -1 is complete, and for sufficiently sharp null support function, it is incomplete. Our results apply also to entire surfaces whose curvature is merely bounded.
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.
The aim of these notes is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.
We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of immersions into non-zero curvature space forms, leading to a model for the transitioning of hypersurfaces among H^n, AdS^n, dS^n and S^n; 3) we prove that a holomorphic family of immersion data corresponds to a holomorphic family of immersions, providing an effective way to construct holomorphic maps into the SO(n,C)-character variety. In particular we will point out a simpler proof of the holomorphicity of the complex landslide; 4) we see how, under certain hypothesis, complex metrics on a surface (i.e. complex bilinear forms of its complexified tangent bundle) of constant curvature -1 correspond to pairs of projective surfaces with the same holonomy. Applying Bers Double Uniformization Theorem to this construction we prove a Uniformization Theorem for complex metrics on a surface.
We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant Gaussian curvature in 2+1 dimensions is isometric to a straight convex domain in the hyperbolic plane.
We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, 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.
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 $K \in [-1,0)$ 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.
This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane are obtained.
We prove that any weakly acausal curve $Γ$ 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 $Γ$ 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 $θ\in(0,π)$, every quasisymmetric homeomorphism of the circle admits a unique extension which is a $θ$-landslide of the hyperbolic plane. These extensions are quasiconformal.
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 $L^1$-energy of maps from $(S,h)$ to $(S,h')$. The study of $L^p$-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 behavior 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.
Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sitter space, where $ρ=(ρ_l,ρ_r)$ is the corresponding representation into $\mathrm{PSL}_2\mathbb R\times \mathrm{PSL}_2\mathbb R$. It is known that, if $σ$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold $Λ_{\mathrm{ML}}$ of $M_ρ$. We show that, given any $ρ$-equivariant embedding $σ$, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$. Conversely, any Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$ is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.
The classical Minkowski problem in Minkowski space asks, for a positive function $ϕ$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $ϕ(η)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $η$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $μ$ 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 $μ$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $Γ$ of isometries of $\mathbb{H}^d$, given a $Γ$ invariant Radon measure $μ$, given a isometry group $Γ_τ$ of Minkowski space, with $Γ$ as linear part, there exists a unique convex set with area measure $μ$, invariant under the action of $Γ_τ$. 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--Ampère equation. The existence part can be translated as an existence result for Monge--Ampère equation. The regular version was proved by T.~Barbot, F.~Béguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $Γ_τ=Γ$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $Γ_τ$-invariant surface of constant Gauss-Kronecker curvature equal to $1$.
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-Ampère equation $\det D^2 u(z)=(1/ψ(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=φ$, for $ψ$ a smooth positive function and $φ$ 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 $φ$ 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 $(-\infty,0)$.
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 Teichmüller 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 $(0,2π)$ 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 Teichmüller 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.
We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions, in which a uniform light signal is emitted on the $r$-level surface of the cosmological time for $r\to 0$. We show that the frequency of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2+1, we show that this observed frequency function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency function for a few simple examples.
We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities of angles less than $2π$ along a time-like graph $Γ$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.
The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichmüller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichmüller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.