Gregory J. Galloway
A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study hypersurfaces with prescribed null expansion in an initial data set. We propose a notion of stability and prove a topology theorem. Eichmair's Perron approach toward the existence of marginally outer trapped surface adapts to the settings of hypersurfaces with prescribed null expansion with only minor modifications.
On the existence of sections with constant surface gravity on null hypersurfaces
本文在满足零收敛条件的时空中确定了一类允许恒定表面引力零截面的自然零超曲面，补充了广义相对论中 Cauchy 视界与黑洞视界相关零膨胀超曲面的已有结果。
Horizons that Gyre and Gimble: A Differential Characterization of Null Hypersurfaces
本文从卡罗尔几何视角，利用带挠率的联络发展零超曲面微积分，以刻画其内在与外在微分不变量，并给出共形不变的例子。
The constraint tensor for null hypersurfaces
本文提出了一种完全基于外蕴几何的零超曲面约束张量定义，证明了其协变性及与时空Ricci张量拉回的一致性，并推广了近视界方程及外蕴曲率重建方法。
Let $M^{n+1}_1$ be a light-like geodesically complete Lorentzian $(n+1)$-manifold satisfying the null energy condition. We show that null hypersurfaces properly immersed in $M^{n+1}_1$ are totally geodesic.
In this paper we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson-Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.
The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of …
In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $Ω$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference background geodesic foliation approaching large spheres and a positive function, constant along the null generators on $Ω$, which describes the relation between the two foliations at infinity. The integrand in the limit expression has interesting covariance and invariance properties wi…
Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$.
We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overlineϕ,ζ, η)$, tangent to the characteristic vector field $ζ$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen conformal and screen totally umbilic null hypersurfaces. We prove that such hypersurfaces are contained in indefinite Sasakian space forms of constant $\overlineϕ$-sectional curvature of $-3$.
We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus provides at any point an elliptic structure intimately connected with the geometry of the null hypersurface, independent of the choice of a specific section through that point. No analytic or algebraic symmetries or other conditions are imposed on the metric. The spectral properties…
In this paper, we obtain the parametric expressions of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on a pseudo null, partially null or null curves in $E^4_1$ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures and mean curvatures. Also, we construct some examples for these canal hypersurfaces and finally, we give some characterizations for tubular hypersurfaces in $E^4_1$.
Null hypersurface data and ambient vector fields: Killing horizons of order zero/one
本文研究带切向零向量场 η 的零超曲面，建立其形变张量与几何张量的恒等式，推广表面引力并提出三类抽象视界，涵盖一般拓扑与不动点情形。
Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riemannian metrics which collapses with bounded curvature and applying known results on the topology of manifolds which collapse. The result is then applied to general relativity, where it implies a restriction on the topology of smooth compact Cauchy horizons in spacetimes with vario…
We show that there is a correspondence between totally umbilic null hypersurfaces in generalized Robertson-Walker spaces and twisted decompositions of the fibre. This allows us to prove that nullcones are the unique totally umbilic null hypersurfaces in the closed Friedmann Cosmological model. We also apply this kind of ideas to static spaces, in particular to Reissner-Nordström and Schwarzschild exterior spacetimes.
The null Penrose inequality, i.e. the Penrose inequality in terms of the Bondi energy, is studied by introducing a funtional on surfaces and studying its properties along a null hypersurface $Ω$ extending to past null infinity. We prove a general Penrose-type inequality which involves the limit at infinity of the Hawking energy along a specific class of geodesic foliations called Geodesic Asymptotic Bondi (GAB), which are shown to always exist. Whenever, this foliation approaches large spheres, this inequality becomes the null Penrose inequality and we recover the results of Ludvigsen-Vickers …
We study the geometry of null hypersurfaces in indefinite complex contact manifolds. We prove several classification results for a variety of well-known null hypersurfaces, including the totally umbilic, totally screen umbilic, and the screen conformal ones. Furthermore, a characterization of the ambient space is given in case the underlying null hypersurface is totally contact umbilic, totally contact screen umbilic or contact screen conformal, i.e. we have proved that the ambient complex contact manifold must be a space of constant $GH$-sectional curvature of $-3$.
We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman-Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstr…
We introduce a class of null hypersurfaces of a semi-Riemannian manifold, namely, screen quasi-conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.
Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $ζ$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $ζ$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.
In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal null 2-type hypersurface in a Euclidean space has constant mean curvature and constant scalar curvature.
Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons
本文建立零超曲面处环境 Ricci 张量横向导数与度规横向展开的协变几何恒等式，并证明非退化 Killing 视界上的完整展开由视界抽象数据及 Ricci 导数塔唯一确定。
Transverse expansion of the metric at null hypersurfaces II. Existence results and application to Killing horizons
本文证明给定零超曲面的完整渐近展开满足约束方程时，存在环境流形无限阶求解 Einstein 方程，并证明任意维度和拓扑的抽象 Killing 视界数据可产生无限阶求解 Λ-真空方程的环境空间。
Null hypersurfaces as wave fronts in Lorentz-Minkowski space
本文证明洛伦兹-闵可夫斯基空间中L-完备零超曲面作为波前可由欧几里得空间超曲面典范诱导，并刻画了奇异集为紧集的此类波前。
In the present paper, we show that the geometry of a screen integrable null hypersurface can be generated from an isometric immersion of a leaf of its screen distribution into the ambient space. We prove, under certain geometric conditions, that such immersions are contained in semi-Euclidean spheres or hyperbolic spaces, and the underlying null hypersurfaces are necessarily umbilic and screen totally umbilic. Where necessary, examples have been given to illustrate the main ideas.
We study the principal configurations around an isolated $η$-umbilical point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of Minkowski space $\mathbb{R}_1^4$ relative to a well-defined null vector field $η$ orthogonal to the surface $S$. In the particular case of $M$ being a null rotation hypersurface of $\mathbb{R}_1^4$ we also recover the local Darbouxian principal configurations around that kind of $η$-umbilical points.
For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $ν:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $ν$ that are both tangent and $g$-orthogonal to $ν.$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $ST^*M$ of an immersed spacelike hypersurface $μ:M^m\to X^{m+1}.$ We show that a Legendrian map $\wt λ: L^{m-1}\to (ST^*M)^{2m-1}$ defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a …
In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel, defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the e…
Riemannian flow techniques on totally geodesic null hypersurfaces
本文研究全测地零超曲面对Lorentzian流形的影响，通过结合rigging技术与零叶状结构证明Riemann流结构的存在性，并探讨曲率对因果结构的限制及周期测地线的存在性。
Rigging technique introduced in \cite{bi0} is a convenient way to address the study of null hypersurfaces. It offers in addition the extra benefit of inducing a Riemannian structure on the null hypersurface which is used to study geometric and topological properties on it. In this paper we develop this technique showing new properties and applications. We first discuss the very existence of the rigging fields under prescribed geometric and topological constraints. We consider the completeness of the induced rigged Riemannian structure. This is potentially important because it allows to use mos…
Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by Klainerman and Nicolò and show that the corresponding geometry is controlled locally only in terms of the initial geometry on $S$ and the $L^2$ curvature flux through $\mathcal{H}$. In particular, we show that the ingoing and outgoing null expansions $\mathrm{tr} χ$ and $\mathrm{tr} \underlineχ$ are both locally uniformly bounded. The proof of our estimates…
The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood -Paley theory, in a noncommutative setting, defined via heat flow on surfaces.
In this work we introduce the notion of constant angle null hypersurface of a Lorentzian manifold with respect to a given ambient vector field. We analyze the case in which the vector field is closed and conformal, thus finding that such null hypersurfaces have a canonical principal direction. We further provide some classification results for constant angle null surfaces with vanishing null mean curvature.
In the present paper, firstly we obtain the general expression of the canal hypersurfaces which are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hypercones in $E_{1}^{4}$ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in $E_{1}^{4}$ by taking constant radius function and finally…
The purpose of the present work is to study (marginally) trapped submanifolds lying in a null hypersurface. Let $(M,g,N)\to\Bm(c)$ be a null hypersurface of a space-time with constant sectional curvature $c$, endowed with a Screen Integrable and Conformal rigging $N$. The (Marginally) Trapped Submanifolds we are interested with are particular leaves of the screen distribution according to the sign of their expansions. We prove that if $c$ is non-positive, then $\Bm$ cannot contain a null non-expanding horizon. In the case $c$ is positive, we show that if $\Bm$ satisfies Einstein's equation and…
Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In par…
Codimension two spacelike submanifolds into null hypersurfaces of generalized Robertson-Walker spacetimes
本文研究广义Robertson-Walker时空中类光锥内的余维二类空子流形，给出径向坐标条件使其分别共形微分同胚于双曲空间、圆圆柱面或球面，并得到弱俘获子流形的非存在性结果。
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide several rigidity results under appro- priate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi-Bernstein type results are given.
We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a ma…
Canal Hypersurfaces Generated by Non-Null Curves with Parallel Frame in Minkowski Space-Time
本文研究闵可夫斯基时空中由具有平行标架的非零曲线生成的管状超曲面，通过计算高斯曲率、平均曲率和主曲率等几何量，给出其参数化一般表达式及重要几何不变量与刻画。
On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes
本文研究一类静态时空中零能量条件对全脐超曲面的影响，分别在类空和类时情形下刻画了常时空平均曲率嵌入曲面与常脐性因子光子曲面。
In recent work, the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, for a class of strictly stable Weakly Isolated Horizons, we show the existence of a unique foliation by quasi-round MOTS. In the second part, we show that any subsequent space-time perturbation continues to admit a quasi-round MOTS. Finally, for perturbations of the quasi-round MO…
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed using level-set methods and an appropriate variational principle. This new flow has a natural application as a variational-type approach to constructing MOTS, and this work also gives new insights into the theory of weak solutions of inverse mean curvature flow.
The main objective of this paper is to control the geometry of null cones with time foliation in Einstein vacuum spacetime under the assumptions of small curvature flux and a weaker condition on the deformation tensor for $\bT$. We establish a series of estimates on Ricci coefficients, which plays a crucial role to prove the improved breakdown criterion in [12].
This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the existence of a ray of maximal length or a maximality condition on the volume of Lorentzian distance balls over the hypersurface. The proof of the first case follows work by Andersson, Galloway and Howard and uses their geometric maximum principle for level sets of the (Lorentzian) B…
Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define for hypersurfaces in $\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r; F}$, a generalization of the usual $r$-th mean curvature function. We call a hypersurface is anisotropic minimal if $H_F=H_{1; F}=0$, and anisotropic $r$-minimal if $H_{r+1; F}=0$. Let $W$ be the set of points which are omitted by the hyperplanes tangent to $M$. We will prove that if an oriented hypersurface $M$ is anisotropic minimal, and the set $W$ is open and non-empty, then $x(M)$ …
Optimal transport on null hypersurfaces and the null energy condition
本文研究 Lorentzian 流形零超曲面上的退化最优传输问题，利用概率测度沿零测地线的 Boltzmann--Shannon 熵凸性刻画零能量条件，并推导出时空稳定性、零锥比较及 Hawking 面积定理等锐利应用。
In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and sufficient conditions are obtained for a hypersurface to be timelike, null, or spacelike in the generalized model; in the non-null case, the geometry is given by a warped product. Several examples of timelike, null, and spacelike hypersurfaces are presented. Lastly, we calculate t…
Horizon Data: Existence Results and a Near-Horizon Equation on General Null Hypersurfaces
本文提出了一种研究一般地平线几何的范式，并推导出广义近地平线方程，证明了地平线存在的两个存在性定理。
Let $x:M\to\Bm$ be the canonical injection of a Null Hypersurface $(M,g)$ in a semi-Riemannian manifold $(\overline{M},\bar g)$. A rigging for $M$ is a vector field $L$ defined on some open set of $\overline{M}$ containing $M$ such that $L_p\notin T_pM$ for each $p\in M$. Such a vector field induces a null rigging $N$. Let $\bar η$ be the 1-form which is $\bar g$-metrically equivalent to $N$ and $η=x^\star\barη$ its pull back on $M$. We introduce and study for a given non vanishing function $α$ on $M$ the so-called $α$-associated (semi-)Riemannian metric $ g_α=g+αη\otimes η$. For a closed rigg…
The paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear waves. We prove a uniform lower bound on the radius of injectivity of these null boundaries in terms of the Riemann curvature flux along them and some additional quantities arising specifically in a problem of a large data breakdown criterion in General Relativity
Spectral splitting theorem and ends of minimal hypersurfaces
本文通过构造加权最小测地线，证明了非负双曲曲率流形中有限指数极小超曲面具有有限端，推广了Li-Wang关于非负截面曲率流形的结果。
Timelike bounce hypersurfaces in charged null dust collapse
研究了带电null尘在坍缩过程中通过静电排斥力反弹的时空结构，构建了包含反弹超曲面的时空模型，并探讨了其正则性和条件性结果。
On the geometry of synthetic null hypersurfaces
本文提出一种合成框架，用于研究非光滑时空中的零超曲面几何与分析特性，定义合成零超曲面并提出合成零能量条件，适用于低正则性时空，并应用于霍金面积定理和彭罗斯奇点定理。
Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime
本文证明了真空微扰Schwarzschild时空中常质量面函数叶化的全局存在性，并通过与球对称叶化比较研究其几何性质，为应用零Penrose不等式奠定核心基础。
In this short paper, we review recent progress on the positive mass theorem for spacelike hypersurfaces which approach to null infinity in asymptotically flat spacetimes. We use it to prove, if the functions $c(u, θ, ψ)$, $d(u, θ, ψ)$ vanish at certain retarded time in vacuum Bondi's radiating spacetimes, then the Bondi mass is nonnegative up to that time.
In this paper, we completely classify the homothetical hypersurfaces having null Gauss-Kronocker curvature in a Euclidean (n+1)-space. Several applications to the production functions in economics are also given.
General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in Gene…
Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces
本文基于临界点理论和谱Ricci界，证明了流形上Δ+V的临界性准则及分裂定理，拓展了Cheeger-Gromoll定理，并研究了非负bi-Ricci或截面曲率流形中稳定和δ-稳定极小超曲面的拓扑性质。
Potential Carroll Structures and Special Carrollian Manifolds
本文研究潜在卡罗尔结构与特殊卡罗尔 manifold 的关系，以描述 null 超曲面的内在几何。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。