The paper studies hyperbolic equations in a spacetime foliation, proving existence and uniqueness.
problem Existence and uniqueness of solutions for first-order linear hyperbolic systems in a double null foliation.
method Proves global existence and uniqueness for first-order linear hyperbolic systems with initial data on a past null hypersurface.
result Derives a novel algebraic constraint for tensorfields satisfying the linearized Bianchi equations.
Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.
The paper defines marginal tubes and proves their null nature.
problem Understanding the geometry of spacelike surfaces in spacetimes.
method Introducing marginal tubes and studying spacelike surfaces with double null coordinates.
result If every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null.
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, f…
New method shows trapped surfaces form in geodesic foliation.
problem Formation of trapped surfaces in Einstein vacuum equation.
method Geodesic foliation, non-integrable PT frame.
result Trapped surfaces form for Einstein vacuum equation.
This paper addresses pure gauge questions in the study of (asymptotically) de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results which 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.
The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
Researchers prove CWY angular momentum is supertranslation invariant in double null gauge.
problem Supertranslation invariance of CWY angular momentum in double null gauge.
method Identified and proved supertranslation ambiguity; showed CWY angular momentum is free of this ambiguity.
result CWY angular momentum is supertranslation invariant in double null gauge.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
Paper defines and proves geometric uniqueness of Einstein field equations.
problem Einstein field equations characteristic Cauchy problem
method Covariant definition of double null data, proving geometric uniqueness
result Double null data fully covariant and geometrically unique
This paper proves a canonical foliation on null infinity for Kerr-like black holes.
problem Establishing well-defined physical quantities on null infinity for Kerr-like black holes.
method Existence and uniqueness results for GCM spheres by Klainerman-Szeftel.
result Existence of a canonical foliation on future null infinity with well-defined physical quantities.
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 pro…
Study on null-projectability of Levi-Civita connections in neutral metrics.
problem Characterizing projectability of Levi-Civita connections along null parallel distributions.
method Analyzing projectability of torsion-free connections along foliations on manifolds, focusing on neutral metric signatures and mid-dimensional distributions.
result Extension of Patterson and Walker's Riemann extension metrics to null parallel distributions of any dimension.
Study on completeness of foliations and null Killing fields in Lorentzian manifolds.
problem Completeness of foliations and null Killing fields in Lorentzian manifolds.
method Analyzes different geometric settings and conditions for completeness, including totally geodesic lightlike foliations and specific affine structures.
result Characterizes completeness for null Killing fields and specific affine structures, and provides non-complete examples.
New method to find surfaces in null cones with constant curvature near black hole indicators.
problem Finding surfaces in null cones with constant curvature near black hole indicators.
method Flow techniques to foliate null cones by surfaces of constant spacetime mean curvature.
result A neighbourhood of stable MOTS in a null cone can be foliated by hypersurfaces of constant spacetime mean curvature.
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 backgr…
The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.
problem Preventing the existence of null geodesic lines in spacetimes.
method Identifying geometric conditions on foliations of spacetimes that prevent null geodesic lines, especially for spacetimes with compact Cauchy hypersurfaces.
result Conditions on foliations can prevent null geodesic lines, leading to restrictions on cosmological spacetime geometry.
Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.
problem Understanding marginally trapped surfaces in perturbed Schwarzschild spacetime.
method Developed a method to study spacelike surfaces in a double null coordinate system.
result For every incoming null hypersurface nearly spherically symmetric, there exists a unique embedded marginally trapped surface.
Using the complex parabolic rotations of holomorphic null curves in C4, we transform minimal surfaces in Euclidean space R3⊂R4 to a family of degenerate minimal surfaces in Euclidean space R4. Applying our deformation to holomorphic null curves in ${…
Study of mean curvature flow on null hypersurfaces leading to MOTS.
problem Detecting marginally outer trapped surfaces (MOTS) in null hypersurfaces.
method Analysis of mean curvature flow on null hypersurfaces with mild conditions.
result Existence and convergence of mean curvature flow to MOTS.
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 Hawki…
In this paper we present a certain class of geodesic vector fields of the double-twisted product R X R. Some examples of totally geodesic foliations are given.
Study linear perturbations in Schwarzschild black hole spacetime.
problem Linear perturbations of Schwarzschild black hole spacetime.
method Investigate linearised perturbation of constant mass aspect function foliation at null infinity.
result Linearised perturbations of Bondi energy and mass vanish, and all linear momentum can be achieved.
Let H denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime (M,g). In this paper we study the so-called canonical foliation on H introduced by Klainerman and Nicolò and show that the corresponding geometry is controlled lo…
We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.
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 pro…
Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.
problem Properties of Lorentzian manifolds influenced by totally geodesic null hypersurfaces.
method Coupling rigging technique with null foliation existence to prove Riemann flow structure.
result Proves curvature conditions restrict causal structure of spacetime.
The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.
problem Understanding null hypersurfaces in Lorentz-Minkowski space.
method Analyzing L-complete null hypersurfaces as wave fronts in Euclidean space. result Most null wave fronts can be realized as restrictions of certain L-complete null wave fronts. A global twistor correspondence is established for neutral self-dual conformal structures with alpha-surface foliation when the structure is close to the standard structure on S^2 times S^2. We need to introduce some singularity for the alpha-surface foliation such that the leaves intersect on a fixed two sphere. In th…
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second the…
The paper extends isometric embedding results to null cones and spheres.
problem Isometric embedding of metrics on spheres in null cones.
method Extending Li-Wang's result to compact manifolds, specializing to 2D, developing existence and uniqueness theorems, and proving foliations.
result Existence and uniqueness of isometric embeddings in null cones.
New findings on L-spaces and taut foliations in hyperbolic links.
problem Characterizing L-spaces and taut foliations in Dehn surgeries on hyperbolic links. method Analyzing rational homology spheres and using coorientable taut foliations.
result Non-meridional surgeries on fibered hyperbolic two-bridge links support coorientable taut foliations.
New proof of Lie algebroid action equivalence and integrability.
problem Equivalence between Lie algebroid action and integrability.
method Double Lie groupoids and multiplicative foliations.
result Simple characterization of double Lie groupoids inducing Lie groupoid structures.
Extends double symplectic groupoids to transitive Courant algebroids.
problem Generalizing double symplectic groupoids to a broader class of Lie bialgebroids.
method Classification of exact twisted Courant algebroids and construction of foliations.
result Generalization of many examples of double symplectic groupoids.
We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse fu…
4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) o…
Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation …
The study connects electromagnetic structures to Legendrian fields on the 3-sphere.
problem Understanding the topology of stable electromagnetic structures.
method Connecting null solutions to Maxwell's equations with Legendrian fields on the 3-sphere.
result Any (possibly knotted) toroidal surface can be realized as a magnetic surface of a null solution, implying stability.
We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild exterior in a slab around a null hypersurface N0. N0 terminates at past null infinity I− and S0:=∂N0…
Study Lie foliation of Walker manifolds in pseudo-Riemannian geometry.
problem Characterize Lie foliations of Walker manifolds.
method Analyzes Walker manifolds with null parallel distributions to integrate to Lie foliations and studies their properties.
result Walker manifolds with null parallel distributions always integrate to Lie foliations, and the transverse holonomy coincides with the image of the holonomy morphism.
In the double field theory, gauge symmetries are realized as generalized diffeomorphisms in the doubled spacetime. By consistency of the theory, dependence of tensor fields on the doubled coordinates is strongly constrained. This causes finite transformation law highly complicated, both technically and conceptually. In…
Geometric quantization for specific symplectic structures proved.
problem Quantization of specific symplectic structures.
method Geometric quantization for constant rank presymplectic structures with Riemannian null foliation.
result Quantization-commutes-with-reduction theorem proved in this context.
Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.
problem Creating foliations of lightcones with specific geometric properties.
method Employing a geometric flow inspired by Huisken-Yau's approach for Riemannian settings.
result Initial data converges exponentially to an STCMC surface under area preserving null mean curvature flow.
A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose M is an oriented 3-manifold with connected boundary a torus, and suppose M contains a properly embedded, compact, oriented, surface R with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…
Study shows contact structure on null geodesic space for specific spacetimes.
problem Understanding contact structures on null geodesic spaces of spacetimes.
method Contactomorphism and embedding techniques in spherical cotangent bundles.
result Space of null geodesics contactomorphic to canonical structure in spherical cotangent bundle.
The paper analyzes null infinity's geometry without restrictions.
problem Understanding null infinity's geometry without constraints.
method Coordinate-free approach, treating conformal factor as dynamical.
result Isometric spacetimes with identical free data at null infinity.
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.