Olaf Müller
In this note a proof is given for global existence and uniqueness of minimal surfaces of Lorentzian type from a cylinder into globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives.
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In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hy…
On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
本文研究带类时边界的整体双曲流形上Friedrichs系统的柯西问题，通过施加容许边界条件证明强解的存在唯一性，且双曲型Friedrichs系统的柯西问题在Hadamard意义下适定，并给出相关实例。
In this short note, a question of patching together globally hyperbolic manifolds is adressed which appeared in the context of the construction of Hadamard states.
Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.
We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.
On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundary
本文研究具有类时边界的全局双曲流形上Faraday张量Cauchy问题的适定性，证明了算子d+δ的Green算子存在性，并构造了解空间上的预辛结构。
The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Einstein's static universe.
Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.
In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.
Enveloping space of a globally hyperbolic conformally flat spacetime
本文通过构造包含所有共形平坦柯西扩张的包络空间，重新证明全局双曲共形平坦时空 C 0-极大扩张的存在唯一性，并证明其保持包含关系。
The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.
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.
In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold $M$ as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the underlying algorithmic construction in the language of ultrastatic Lorentzian manifolds. Subsequently we show that the construction carries over to the case of static backgrounds thanks to a suitable reduction to the ultrastatic scenario. Finally we prove that the overall procedure …
We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological constant.
We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences) of globally hyperbolic space-times. In the same setting we indicate as an application the extension of a previous result on the Cauchy problem for the wave equation.
We give an example of a spacetime with a continuous metric which is globally hyperbolic and exhibits causal bubbling. The metric moreover splits orthogonally into a timelike and a spacelike part. We discuss our example in the context of energy conditions and the recently introduced synthetic timelike curvature-dimension (TCD) condition. In particular we observe that the TCD-condition does not, by itself, prevent causal bubbling.
Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauchy hypersurface.
Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces
针对整体双曲洛伦兹长度空间，证明了合成邦内-迈尔斯刚性定理：曲率下界为K<0且存在特定长度开距离实现者时，空间必为带cos翘曲函数的翘曲乘积，还推导出曲率上界为K<0的整体双曲时空的同类结论。
We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kähler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.
Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.
Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes
本文证明了 globally 超曲率最大 conformally 平面时空中的点的未来集在特定条件下具有极大性，并提供了新证明。
Refocusing spacetimes need not be strongly refocusing
本文证明了存在全局双曲的重新聚焦时空但不是强重新聚焦的，并回答了Chernov、Kinlaw和Sadykov的问题，同时证明了Legendre重新聚焦的全局双曲时空允许强重新聚焦度量。
Let $ψ$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $ψ$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980's. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigat…
It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.
In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fiber compact.
On complete trapped submanifolds in globally hyperbolic spacetimes
本文研究全局双曲时空中抛物型类空子流形的刚性与不存在性，通过因果平均曲率向量场条件，并应用于广义相对论中的陷俘曲面，恢复并推广了标准静态及广义罗伯逊-沃克时空中的相关结果。
Let $S$ be a closed oriented surface of genus at least $2$. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over the Teichmüller space of $S$, we study the behaviour of these geometric structures along pinching sequences. We show, in particular, that the regular globally hyperbolic anti-de Sitter structures introduced in arXiv:1806.08176 naturally appear as limiting points.
Lipschitz continuity of the cut time for globally hyperbolic spacetimes
研究全局双曲时空固定点相关的聚焦时间和割线时间的性质，采用伊藤 - 田中的方法并适用于洛伦兹设置，证明了它们的局部利普希茨连续性，推导相关估计和准则，得出割迹的豪斯多夫余维数结论。
A closure result for globally hyperbolic spacetimes
该研究借助一般性Bonnet-Myers型结果，证明了满足某一时刻关于减速、压强、哈勃常数自然假设的全局双曲时空的闭包结果，丰富了微分几何与广义相对论相关时空性质研究。
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.
We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero mode…
Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime
本文研究全局双曲时空上代价为 $c_2$ 的最优输运问题，通过分析弱 Kantorovich 势（$\pi$-解）的正则性，证明两 Borel 概率测度间最优输运映射的存在性、唯一性与结构。
In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semidirect product between $\R^{n-2}$ and one of $\1, SU(k), Sp(1), G_2$ and $Spin(7)$.
On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime
本研究拓展前人工作，探究整体双曲时空中多条最大化测地线轨迹的拓扑性质，证明其局部可缩，定义洛伦兹Aubry集并证明相关包含关系为同伦等价。
In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of $C^*$-algebras. We demons…
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).
We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheimer-Penrose causal completion of M, retaining essentially all the good properties of previous topologies in this ambient. In particular, we show that if the globally hyperbolic spacetime $M$ admits a conformal boundary, defined in such broad terms as to include all the standard exa…
Minimizing the volume of globally hyperbolic anti-de Sitter 3-manifolds
本文证明最大全局双曲Cauchy紧致反德西特3-流形M的体积至少为π²|χ(M)|，且该最小值当且仅当M是Fuchsian时达到。
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
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.
The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal extension. Since the causal structure of a space-time is invariant under conformal changes of metrics we may generalize this notion of maximality to the conformal setting. In this article we focus on conformally flat space-times of dimension greater or equal than 3. In this case…
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.
Causal completion of a globally hyperbolic conformally flat spacetime
本文研究全局双曲时空的因果完备化，证明在单连通且共形平坦条件下其为同胚于S×[0,1]的带边拓扑流形，并刻画三类时空族的完备化。
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic s…
Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the…
The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
该研究针对含完整类时线、非负类时曲率的全局双曲洛伦兹长度空间，通过构造类时渐近线、引入类时线平行性概念，证明了综合分裂定理，并将局部分裂结果推广至全局。
Uniqueness of Hypersurfaces of Constant Higher Order Mean Curvature in Hyperbolic Space
本文研究了双曲空间中极球面与等距球面的唯一性，将 Bernstein 定理推广至常高阶平均曲率的嵌入超曲面，证明了浸入超曲面的 Bernstein 型结果，并确立了极球面与等距球面的刚性。
We investigate the equation $$(-Δ_{\mathbb H^n})^γ w=f(w)\quad in \mathbb H^{n},$$ where $(-Δ_{\mathbb H^n})^γ$ corresponds to the fractional Laplacian on hyperbolic space for $γ\in (0,1)$ and $f$ is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to $\pm 1$ at any point of the two hemispheres $S_\pm \subset \partial_\infty \mathbb H^n$ and which is strictly increasing with respect to the signed distance to a tot…
Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $\partial M$, the splitting of any globally hyperbolic $(\overline{M},g)$ as an orthogonal product ${\mathbb R}\times \barΣ$ with Cauchy slices with boundary $\{t\}\times \barΣ$ is pro…
The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= \R \times {\cal S}$, $g= - β(τ,x) dτ^2 + \bar g_τ$, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $τ$ with timelike gradient $\nabla τ$ on all $M$.
Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points
本文在无共轭点且基本群为Gromov双曲、剩余有限的闭黎曼流形上，通过对扩张集施加适当条件，证明了测地流最大熵测度的唯一性，从而推广了此前仅在负截面曲率背景度量下成立的相关结果。
We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationary. This gives certain rigidity or uniqueness results for exact AdS and related spacetimes.
The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this funct…
We study the number of solutions of the asymptotic Plateau problem in H^3. By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of C^3 Jordan curves in S^2_{infty}(H^3) such that any curve in this set bounds a unique least area plane in H^3.
Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space
该研究针对三维双曲空间中的渐近Plateau问题，证明了解的唯一性等价于稳定极小圆盘类中的唯一性，给出了满足唯一性的曲线条件，并构造出存在不可数多稳定极小圆盘解的拟圆周实例。
We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $χ$ such that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic. (i…
Degeneration of globally hyperbolic maximal anti-de Sitter structures along rays
本文借助 Teichmüller 空间余切丛参数化，分析 GHMC anti-de Sitter 结构沿二次微分射线退化时 Hausdorff 维数、凸核宽度与 Hölder 指数等几何量的行为。
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。
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…
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