Do Link Polynomials Detect Causality In Globally Hyperbolic Spacetimes?

Legendrian links, causality, and the Low conjecture

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Natário and Tod says that two events $x,y\inß$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological l…

Linking and causality in globally hyperbolic spacetimes

The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbolic spacetime $(X^{m+1}, g)$. The spherical cotangent bundle $ST^*M$ is identified with the space $N$ of all null geodesics in $(X,g).$ Hence the set of null geodesics passing through a point $x\in X$ gives an embedded $(m-1)$-sphere $S_x$ in $N=ST^*M$ called the sky of $x.$ Low o…

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.

用辛Quandle检测因果性

Detecting Causality with Symplectic Quandles

本文证明辛Quandle结合Alexander-Conway多项式可区分Allen-Swenberg 2-sky类链环与两个Hopf链环的连通和，从而为检测(2+1)维全局双曲时空中的因果性提供了新的拓扑不变量工具。

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).

Wild globally hyperbolic maximal anti-de Sitter structures

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned hyperbolic surfaces and as the bundle over the Teichmüller space of $Σ$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.

Regular globally hyperbolic maximal anti-de Sitter structures

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $Σ$ and as the bundle over the Teichmüller space of $Σ$ whose fibre consists of meromorphic quadratic differentials with poles of order at most $2$ at the punctures.

Alexander Quandles and Detecting Causality

In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes this, while the Alexander-Conway polynomial, independently, is insufficient. As the Alexander-Conway polynomial on its own is not likely to detect causality an additional piece of information must be supplemented to the link polynomial. This paper aims to examine the ability of Alexander quandles, to distinguish the connected sum of two Hopf links and Allen-…

高签名下的全局超曲性

Global hyperbolicity in higher signatures

本文研究了高签名下伪黎曼流形的全局超曲性，推广了该概念并证明了因果钻石的紧致性，同时推导了Plateau问题解的存在性及GH-正则表示的特征化。

On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds

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 corol…

Fenchel-Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel-Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.

medial 拟群检测因果性的能力及其对特定链环和纽结的着色性质

On Medial Quandle and Detecting Causality

研究medial拟群检测（2+1）维整体双曲时空因果性的能力，证明其在特定等价关系下无法区分关键链环（含Alexander拟群实例），并推导其对特定缠结着色的广义定理以助力区分更广泛纽结链环。

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。

平坦(2+1)-时空中的多面体曲面与双曲曲面上的平衡胞腔剖分

Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

本文证明了负奇点曲率的平坦度量可等距嵌入为平坦(2+1)-时空中的凸多面体Cauchy曲面，并通过Gauss映射将其转化为双曲曲面上的平衡测地胞腔剖分问题，基于Teichmüller空间总长度的凸性证明了唯一性。