Connection by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field

Lightlike sets with applications to the rigidity of null geodesic incompleteness

An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified concusions may arise, showing that those conclusions will fail only in special cases, at least some of which may be described. These are the so-called rigidity theorems, and have many important examples in the especialized literature. In this paper, we prove rigidity results for generalized plane waves and certain globally hyperbolic spacetimes in the prese…

Anosov representations as holonomies of globally hyperbolic spatially compact conformally flat spacetimes

Anosov representations were introduced by F. Labourie [18] for fundamental groups of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into real semisimple Lie groups. In this paper, we focus on Anosov representations into the identity component O0(2, n) of O(2, n) for n $\ge$ 2. Our main result is that any Anosov representation with negative limit set as defined in [8] is the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the…

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (charac…

低维中的复广义spin-c Killing旋子

Complex generalized spin-c Killing spinors in low dimensions

该文分类了低维黎曼流形上带特定复Killing函数的复广义spin-c Killing旋子，证明了相关猜想不成立，补充了反例并明确了三维情形的等价条件。

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standard timelike Killing vector field by using Morse theory on the associated Finsler manifold. To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike…

Rigidity in vacuum under conformal symmetry

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.

紧致柯西视界具有恒定表面引力

Compact Cauchy horizons admit constant surface gravity

本文证明，在任意时空维度和零能量条件下，每个完全测地连通光滑紧致零超曲面（即紧致柯西视界）都存在一个光滑的类光切向量场，其表面引力为常数。

Jacobi fields on normal homogeneous Riemannian manifolds

We prove that a normal homogeneous space with the property that every Jacobi field along a geodesic vanishing at two points is the restriction of a Killing field along that geodesic is a globally symmetric space.

Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups thes…

黎曼与洛伦兹三维流形上的 Killing 向量场

Killing vector fields on Riemannian and Lorentzian 3-manifolds

本文完整局部分类了具有非零Killing向量场的黎曼与洛伦兹三维流形，推广了Sasakian结构，并给出局部共形平坦及全局完备性和常正曲率度量的条件。

利用同伦理论研究仿射控制系统解以获得 Randers-Kropina 度量的多条连接测地线

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

本文借助 Lusternik-Schnirelman 理论及受控漂移仿射控制系统解空间的同伦结果，证明不可缩流形上任意两点间存在无穷多条 Randers-Kropina 测地线。

具有常非零表面引力的紧致 Cauchy 视界的真空时空的对称性

Symmetries of vacuum spacetimes with a compact Cauchy horizon of constant non-zero surface gravity

本文证明含可归一化为非零常数表面引力的紧致 Cauchy 视界的光滑真空时空必存在 Killing 向量场，支持强宇宙监督猜想。

风 Finsler 结构：从 Zermelo 导航到时空因果性

Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes

本文提出风 Finsler 结构，统一了 Zermelo 导航问题与具有非零 Killing 向量场的时空因果结构描述，并给出导航解、度量性质及一般 Fermat 原理等应用。

Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.

Pseudo focal points along Lorentzian geodesics and Morse index

Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jaco…

Wave equations with initial data on compact Cauchy horizons

We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant. Examples include the Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles. Using this estimate we prove that if a linear wave equation can be s…

零测地线旅行时间的共形边界刚性

Conformal boundary rigidity from null geodesic travel times

研究渐近反德西特时空中零测地线从共形无穷远出发并返回的聚焦性质，证明若所有从同一点出发的测地线在无穷远对极点重聚，则时空共形于反德西特空间，并给出在零能量条件、静态或全局平稳情形下的证明。

沿类时测地线粘合小黑洞 I：形式解

Gluing small black holes along timelike geodesics I: formal solution

本研究在满足爱因斯坦真空方程的(3+1)维整体双曲时空中，沿类时测地线构造了一族小参数依赖的近似解度量，验证了小黑洞引力自作用力研究的形式微扰框架合理性，可用于模拟极端质量比并合。

沿类时测地线粘合小黑洞 III：真实解的构造与极端质量比并合

Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers

该研究作为系列工作的第三部分，通过修正近似解构造出沿类时测地线运动的小质量Kerr黑洞的爱因斯坦方程真实解，并将其应用于极端质量比黑洞并合的时空建模。