Perturbative solutions to the extended constant scalar curvature equations on asymptotically hyperbolic manifolds
Erwann Delay
本文证明了A.布彻，在渐近双曲线设置。这给出了具有常数量曲率的渐近双曲度量的一个新的局部构造。
We prove local existence of solutions to the extended constant scalar curvature equations introduced by A. Butscher, in the asymptotically hyperbolic setting. This gives a new local construction of asymptotically hyperbolic metrics with constant scalar curvature.
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A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners Along a Hypersurface
本文采用类似于[M]的方法建立了允许角点沿超曲面的渐近双曲自旋流形的正质量定理。主要的分析使用的扰动本征函数方程的解的积分表示，以获得在正确的顺序的解决方案的渐近展开。这使我们能够理解渐近双曲度规的共形变化的质量方面的变化。
Asymptotically Hyperbolic Metrics on Unit Ball Admitting Multiple Horizons
本文在单位球$\mathbf{D}^3$上构造了一个数量曲率为-6的渐近双曲度量，该度量包含多个视界.
Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces
本文研究了具有两个渐近双曲端点的Liouville曲面上的逆散射问题。Liouville曲面的主要性质在于测地线流的Hamilton-Jacobi方程的完全可分性。一个重要的相关结果是，驻波方程可以分为一个系统的径向和角常微分方程。渐近双曲Liouville曲面上标量波动方程的固定能量全散射矩阵可以通过考虑其对角分离常微分方程对应的广义调和函数的限制而得到简化。得到的部分散射矩阵包括在一个可数的一组$2 \times 2$矩阵，其系数是所谓的传输和反射系数。结果表明，对于以广义角动量为谱参数的径向常微分方程，反射系数是广义Weyl-Titchmarsh函数。利用复角动量方法和最近关于广义Weyl-Titchmarsh函数的一维反问题的结果，我们证明了在固定的非零能量下反射算子的知识足以唯一地确定所考虑的渐近双曲Liouville曲面的度量.
We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenever a geometrically motivated system of elliptic equations admits a solution. The inequalities treated here relate mass, angular momentum, charge, and horizon area.
Asymptotically hyperbolic metrics on the unit ball with horizons
本文构造了一族具有视界且数量曲率等于-6的渐近双曲流形。我们构造的流形可以任意接近无穷远处的反德西特-史瓦西流形。因此，流形的质量可以很大也可以很小。本文的主要论点是在\cite{M}中使用的胶合方法。
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spaces
本文为渐近双曲Einstein流形中由Graham-Witten极小曲面与共形无穷远界定的区域定义了重整化体积，证明了其Gauss-Bonnet定理并计算了极小超曲面变分下的导数。
Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds
证明了渐近双曲空间上拉普拉斯算子的预解式以有限秩极点亚纯扩张到复平面当且仅当度量是“偶”的（在某种意义上）.如果它不是偶数，则存在一些情况，其中预解式在非物理层中具有本质奇点。
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
Existence and Uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds
本文证明了具有正质量的Anti-de Sitter-Schwarzschild度量渐近的3-流形中具有常中曲率的稳定球面的叶理的存在唯一性。这些度规自然地作为具有负宇宙学常数的爱因斯坦方程的解的类空时间片出现。
Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with compact boundary, we prove the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a fun…
The mass of asymptotically hyperbolic Riemannian manifolds
我们提出了一组全局不变量，称为“质量积分”，它可以定义为一个大类的渐近双曲黎曼流形。当“边界在无穷大”有球形拓扑结构的一个单一的不变量是获得，所谓的质量，我们显示其积极性。我们应用共形紧流形的定义，并证明了质量是完备独立的。我们还证明了结果，密切相关的问题在手，共形紧流形的共形完备是唯一的。
We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.
We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geodesic flow is non-trapping, without having to construct a parametrix. We prove that the size of the strip to which the resolvent can be extended meromorphically is the same obtained by Guillarmou.
The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes
时间片公理指出，可以在任意小的时间间隔内测量的可观测量足以预测所有其他可观测量。虽然众所周知的自由场论的有效性的时间片公理是一个直接后果的领域方程，它是不知道是否也持有一般的相互作用理论，唯一的例外是某些superrenormalizable模型在2维。本文证明了时间片公理至少对形式重整化微扰论中的标量场成立。
Boundary Rigidity in a fixed conformal class for Asymptotically Hyperbolic Manifolds
研究渐近双曲流形中边界距离决定度量的问题，证明若两个共形渐近双曲度量（简单或负曲率）的标记重正化边界距离在共形无穷的某些代表下一致，则两度量相等。
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates
本文给出了作者最近提出的一种分析渐近双曲空间上拉普拉斯算子的新方法。这种新方法特别构造了偶度量（在Guillarmou意义下）预解式的解析延拓，并给出了高能量的条估计。关键的想法是一个扩展的边界上的一个问题，从拉普拉斯变换的频谱参数。扩展的问题是非椭圆的-事实上，在另一方面，它与渐近de Sitter空间上的Klein-Gordon方程有关-但尽管如此，它可以通过Fredholm理论的方法进行分析。这种方法是分析偏微分方程的更一般方法的一个特例，其中包括例如Kerr-de Sitter和Minkowski型空间。本文件是独立的，并处理渐近双曲空间没有负担的读者只需要的材料分析的洛伦兹问题考虑在作者的原始工作。
Hyperbolic formulations and numerical relativity II: Asymptotically constrained systems of the Einstein equations
本文研究了渐近约束系统的爱因斯坦方程的数值积分问题，该问题的目的是使初始数据的自由演化对扰动误差具有鲁棒性。首先，我们研究了以前提出的“$λ$ -系统”，它引入了人工流的基础上的对称双曲制定的约束表面。我们表明，这个系统的工作原理预期的波的传播问题，在麦克斯韦系统和广义相对论使用Ashtekar的连接制定。其次，我们提出了一种新的控制稳定性的机制，我们称之为“调整系统”。通过在动力学方程中加入约束项并调整其乘子，可以简单地得到这一结果。我们解释了为什么一个特定的选择乘数减少了数值误差从非正或纯虚的调整约束传播方程的特征值。这种"调整系统”也在麦克斯韦系统和Ashtekar系统中进行了测试。这种机制影响的不仅仅是系统的对称双曲性。
On Asymptotically Almost Periodic Solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic Manifolds
本文研究实双曲流形上抛物-椭圆Keller-Segel系统，利用热半群估计和不动点论证证明渐近概周期解的存在唯一性及指数衰减。
We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes.
In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{ä}ckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the Helmholtz equation can be separated into a system of a radial ODE and two angular ODEs. We can thus decompose the full scattering operator onto generalized harmonics and the resulting partial scattering matrices consist in a countable set of $2 \times 2$ matrices whose coefficients are the so-called transmission and reflection coefficients. It is shown that t…
The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the Dirac wave functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice …
Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold
本文研究实双曲流形上向量抛物型方程的渐近概周期温和解，基于半群色散光滑估计与不动点论证证明线性及半线性方程解的相关性质，结果可应用于Navier-Stokes方程等。
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