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

Constant mean curvature foliation of globally hyperbolic (2+1)-spacetimes with particles

Let $M$ be a globally hyperbolic maximal compact $3$-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that $M$ admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularities of angles less than $π$ along time-like geodesics).

Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compact AdS 3-manifolds

We consider globally hyperbolic maximal anti de Sitter 3-manifolds $M$ with a closed Cauchy surface $S$ of genus greater than one and prove that any pair of hyperbolic metrics on $S$ can be realized as the boundary metrics of the convex core of a maximal globally hyperbolic anti de Sitter 3-manifold structure on $M$. This answers the existence part of a question of Mess about the unique realization of such metrics. Our theorem has a nice formulation purely in terms of 2-dimensional Teichmüller theory and earthquakes.

Volume maximization and the extended hyperbolic space

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space -- the natural extension of hyperbolic space by the de Sitter space -- except for the degenerate case where all simplices are Euclidean in a generalized sense. Those extended hyperbolic structures can realize geometrically a dec…

关于闭反德西特3-流形的Tholozan体积公式

On Tholozan's volume formula for closed anti-de-Sitter 3-manifolds

本文利用Chern-Simons理论为闭反德西特3-流形的Tholozan体积公式提供了一种新的证明。

Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3…

双曲和反德西特空间中的嵌入凸曲面

Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

给定拟圆或约当曲线及满足曲率条件的度量，构造光滑等距嵌入到双曲或反德西特空间，并连续延拓至边界。

最大全局双曲反德西特三维流形形变空间的para-hyperKähler几何

Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

本文研究闭曲面Σ上MGHC反德西特三维流形形变空间的para-hyperKähler几何，证明其具有中性度量、三个辛结构、复结构与两个para-复结构，并揭示其与Mess同胚、K-曲面参数化及特征簇几何的深刻联系。

Teichmüller 与反德西特几何中的长度函数

Length functions in Teichmüller and anti de Sitter geometry

本文建立了 Teichmüller 空间长度函数与反德西特 3-流形几何的联系，并以此给出长度函数凸性与二阶变分界的新证明，同时引入剪切-弯曲坐标。

非Fuchs表示之间的支配关系与反de Sitter几何

Domination between non-Fuchsian representations and anti-de Sitter geometry

本文受曲面群表示相关研究启发，研究闭曲面群非Fuchs表示间的新支配问题，对容许分支调和浸入的表示求解该问题，证明其余情形不可普遍求解，还关联到带奇点反de Sitter 3流形并构造多族相关分支流形。

双曲空间与反德西特空间中完备曲面的Weyl问题

On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces

本文提出双曲空间及反德西特空间中Weyl问题及其对偶的三个猜想性推广，涉及薄/厚渐近边界的凸域和完备凸曲面，与Kleinian群、Koebe圆域猜想及圆填充等当代几何问题相关，并收集部分新结果。

Crooked surfaces and anti-de Sitter geometry

Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Guéritaud-Kassel lift to restrictions of crooked surfaces in Einstein space …

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

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 c…

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

Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of …

Maximal surfaces in anti-de Sitter 3-manifolds with particles

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $π$. We interpret this result in terms of Teichmüller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than $π$.

具有凸边界的反德西特(2+1)时空在Fuchsian轨迹附近的刚性

Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus

本文证明了在Fuchsian轨迹附近，具有严格凸类空边界（光滑或多面体）的整体双曲紧致反德西特(2+1)时空由边界诱导度量唯一确定。