New vacuum spacetimes without CMC Cauchy surfaces found.
problem Finding vacuum cosmological spacetimes without CMC Cauchy surfaces.
method Extended construction of [6] using spatial topologies M#M. result Obtained a large class of vacuum cosmological spacetimes.
Paper glues characteristic data to Kerr spacetime, proving spacelike gluing.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Detailed characteristic gluing of strongly asymptotically flat data to Kerr spacetime.
result Alternative proof of spacelike gluing construction for strongly asymptotically flat spacelike initial data.
Paper constructs non-symmetric collapsing spacetimes without symmetries.
problem Forming non-symmetric collapsing spacetimes in vacuum.
method Modified Christodoulou's a priori estimates and gluing construction.
result Past geodesic completeness and asymptotic Minkowski space.
The paper extends gluing theorems for linearized gravitational fields in static spacetimes with cosmological constant.
problem Establishing gluing theorems for linearized vacuum gravitational fields on characteristic surfaces.
method Analyzing linearised Einstein equations in Bondi gauge on static four-dimensional spacetimes with cosmological constant.
result Generalization and extension of gluing theorems to include cosmological constant and arbitrary topology.
In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. In particular, we state conditions for the existence of outer tr…
Constructs solutions of Einstein equations for black holes gluing along timelike geodesics.
problem Constructing solutions of Einstein equations for black holes gluing along timelike geodesics.
method Constructs solutions gε of the Einstein equations describing a mass ε Kerr black hole traveling along a timelike geodesic C. result Constructs true solutions gε of the Einstein equations for black holes gluing along timelike geodesics. Study gluing event horizons of Minkowski and Schwarzschild spacetimes.
problem Gluing event horizons of different spacetimes.
method Nonperturbative, gluing results from Aretakis-Czimek-Rodnianski and Christodoulou's short pulse method.
result Construct examples of vacuum gravitational collapse to very slowly rotating Kerr black holes.
The paper extends gluing theorems for gravitational fields in higher dimensions.
problem Proving gluing theorems for linearised gravitational fields on characteristic hypersurfaces.
method Analyzing linearised vacuum gravitational fields in (n+1)-dimensional static spacetimes with cosmological constant. result Generalization of gluing theorems to higher dimensions, extending previous work on light cones.
New method glues Lorentzian spaces, preserving curvature bounds.
problem Creating new spaces from existing ones in Lorentzian geometry.
method Introducing an amalgamation process for Lorentzian pre-length spaces.
result Gluing preserves upper curvature bounds in spacetimes.
New cosmological spacetimes without CMC Cauchy surfaces found.
problem Finding CMC Cauchy surfaces in cosmological spacetimes.
method Generalized Bartnik's construction to connected sums of three-manifolds.
result Cosmological spacetimes without CMC Cauchy surfaces for any compact three-manifolds.
Paper solves Einstein vacuum equations gluing problem with applications.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Codimension-10 gluing construction, relating charges to ADM energy, and localized version.
result Asymptotically flat data can be glued to Kerr spacetime with 10 charges.
Constructs many black hole spacetimes in de Sitter space.
problem Creating well-controlled many black hole spacetimes in de Sitter space.
method Gluing Schwarzschild-de Sitter or Kerr-de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions.
result Solves the Einstein equation directly for the metric, given scattering data at the future conformal boundary.
Researchers prove a nonlinear gluing theorem for gravitational fields near static backgrounds.
problem Proving a nonlinear gluing theorem for gravitational fields near static backgrounds.
method Proved a nonlinear characteristic Ck-gluing theorem for vacuum gravitational fields in Bondi gauge. result Generalized the C2-gluing theorem near light cones to a wider class of hypersurfaces. Formally constructs metrics near timelike geodesics in vacuum spacetimes.
problem Constructing metrics near timelike geodesics in spacetimes.
method Constructs a family of metrics depending on a small parameter ε, solving the Einstein vacuum equations modulo O(ε^∞).
result The rescalings near the geodesic tend to a fixed subextremal Kerr metric.
New spacetimes found that are refocusing but not strongly refocusing.
problem Characterizing spacetimes that are refocusing but not strongly refocusing.
method Analyzing globally hyperbolic spacetimes and introducing new refocusing concepts.
result Existence of spacetimes that are refocusing but not strongly refocusing.
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 …
Chruściel, Isenberg, and Pollack constructed a class of vacuum cosmological spacetimes that do not admit Cauchy surfaces with constant mean curvature. We prove that, for sufficiently large values of the gluing parameter, these examples are both future and past null geodesically incomplete. The authors are honored to de…
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 stationar…
Proves closure for specific spacetimes with certain conditions.
problem Proving closure for globally hyperbolic spacetimes.
method Using a Bonnet-Myers type result.
result Proves closure for spacetimes with specific conditions.
Khovanov homology detects causality in spacetimes.
problem Detecting causality in spacetimes.
method Khovanov homology applied to (2+1)-dimensional spacetimes. result Khovanov homology detects causality in specific spacetimes.
Global Double Field Theory is a higher-dimensional generalization of Kaluza-Klein theory.
problem Formulating a global theory for higher-dimensional gauge fields.
method Generalizing Kaluza-Klein theory to higher principal bundles and higher gauge fields.
result Higher Kaluza-Klein geometry provides a global formulation for Double Field Theory.
We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. Fo…
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
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 con…
Study causal structure of warped spacetimes using novel pre-length spaces.
problem Understanding the causal structure of warped spacetimes.
method Novel notion of Lorentzian pre-length spaces and proof of causal completion as globally hyperbolic pre-length space.
result Causal completion of GRW spacetime is a globally hyperbolic pre-length space under Hausdorff chronological topology.
This paper completes globally hyperbolic conformally flat spacetimes, proving they are topological manifolds.
problem Understanding the structure of spacetimes with specific properties.
method Analyzing globally hyperbolic conformally flat spacetimes, proving their causal completions are topological manifolds.
result Causal completions of globally hyperbolic conformally flat spacetimes are topological manifolds homeomorphic to S x [0, 1].
Stability proved for open Milne spacetime, showing gravity's long-term behavior.
problem Global stability of open Milne spacetime for Einstein-scalar field equations.
method Gaussian normal coordinates, exploiting expanding geometry of Milne spacetime.
result Spatial metric tends to hyperbolic metric as time goes to infinity.
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 Cauc…
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of desc…
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
Maximal spacetimes have unique past/future sets.
problem Characterizing maximal spacetimes.
method Developing map analysis and diamond properties.
result Maximal spacetimes have unique past/future sets.
Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedd…
The chapter explores globally hyperbolic spacetimes using topology and functional analysis.
problem Understanding global hyperbolicity in spacetimes.
method Foundational tools from order theory and topology, geometric analysis, and connections to physics.
result A connection between global hyperbolicity and geodesic completeness of space-like surfaces.
Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.
problem Understanding causal bubbling in spacetimes.
method Example of a globally hyperbolic spacetime with a continuous metric, splitting orthogonally into timelike and spacelike parts.
result The synthetic timelike curvature-dimension (TCD) condition does not prevent causal bubbling.
Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Ca…
Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension n≥3 with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…
Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.
problem Null Penrose inequality on a null hypersurface.
method Global existence of constant mass aspect function foliation on a nearly spherically symmetric incoming null hypersurface in a vacuum perturbed Schwarzschild spacetime.
result Geometry of the constant mass aspect function foliation compared to the spherically symmetric foliation in the Schwarzschild spacetime.
Wave propagator constructed on globally hyperbolic spacetimes.
problem Wave propagation on globally hyperbolic spacetimes.
method Reinterpretation of wave propagator construction on ultrastatic Lorentzian manifolds, reduction to static backgrounds, generalization to globally hyperbolic spacetimes.
result Global wave propagator constructed on globally hyperbolic spacetimes.
Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a global structure of a (compactified) moduli space of Yang--Mills instantons, and a…
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 Cauc…
Study proves existence of global solutions for Standard Model on expanding spacetimes.
problem Existence of global solutions for the Standard Model on expanding spacetimes.
method Gauge-invariant energy estimate for the Euler-Lagrange equations.
result Existence of global solutions for the Standard Model under specific conditions.
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 singularit…
For 2+1 spacetime dimensions, we derive sufficient conditions for the twisting function in a twisted product spacetime, such that there is a global foliation by spacelike CMC surfaces.
Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.
problem Rigidity and non-existence of trapped submanifolds in spacetimes.
method Analysis of parabolic spacelike submanifolds with causal mean curvature in orthogonally splitted and globally hyperbolic spacetimes.
result Non-existence of local minima or maxima of certain functions on submanifolds, leading to rigidity results.