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
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.
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 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.
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.
Solves characteristic problem in general relativity for null data.
problem Characteristic Cauchy problem in Einstein vacuum field equations.
method Abstract data formalism and tangential components of the ambient Ricci tensor.
result Formulates and solves the characteristic problem completely abstractly.
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…
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…
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.
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…
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.
New homologies defined for null homologous links in RP^3, linking to Heegaard Floer homology.
problem Khovanov-type homologies for null homologous links in RP3. method Defined Khovanov-type homologies with input α consisting of graded vector spaces and maps. result Spectral sequence from new homology theory converges to Heegaard Floer homology of even branched double cover.
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rn into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an (n−2)-dimensional plane at 120∘ …
Proves stability of Schwarzschild black holes without symmetry assumptions.
problem Stability of Schwarzschild black holes under general conditions.
method Teleologically normalised double null gauges, analysis of linear stability, and control of non-linearities.
result Proves non-linear asymptotic stability of Schwarzschild family as solutions to Einstein vacuum equations.
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…
Researchers extend microlocal analysis across event horizons of rotating black holes.
problem Incomplete microlocal theory of fields across black hole event horizons.
method Extended microlocal theory for extremal rotating black holes, showing null covectors form an involutive double characteristic manifold.
result Mathematical basis for asymptotic oscillatory solutions near event horizons.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of…
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.
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.
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.
In the 60's Levine proved that if R is a slice knot, then on any genus g Seifert surface for R there is a g component link J, called a derivative of R, on which the Seifert form vanishes. Many subsequent obstructions to R being slice are given in terms of slice obstructions of J. Many of these obstructi…
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. Thi…
Solves C^3 null gluing problem for Einstein vacuum equations.
problem Null gluing of up to third-order derivatives of the metric in Einstein vacuum equations.
method Linear and nonlinear analysis of characteristic data close to Minkowski data.
result Solvable up to a 20-dimensional space of obstructions, 10 of which are novel.
Study on slicing knots in 4-manifolds, focusing on CP^2-slicing numbers.
problem Understanding the slicing properties of knots in 4-manifolds.
method Lower and upper bounds on CP^2-slicing numbers using double branched covers and Seifert forms.
result Findings on the finite and distinct CP^2-slicing numbers for certain knots.
Develops support theorem for analytic transforms in tomography.
problem Analytic wave front set resolution for integral transforms.
method Microlocal analysis, double fibration framework, wave packet transforms.
result Uniqueness and support theorems for analytic transforms.
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.
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
Unique global solutions found for specific initial data.
problem Einstein-scalar-field equations with specific initial conditions.
method Spherically symmetric analysis of small, slowly decaying data.
result Unique global solutions exist for the equations.
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rn into q cells of prescribed (positive) Gaussian measure when 2≤q≤n+1, is to use a "simplicial cluster", obtained from the Voronoi cells of q equidistant points. Moreover, we prove that…
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
Defines constraint tensor for null hypersurfaces, providing explicit geometry.
problem Defining constraint tensor for null hypersurfaces with any topology.
method Explicit definition in extrinsic geometry, covariant for any topology.
result Simple form of constraint tensor on transverse submanifolds.
Compact ECS manifolds have a simple topological structure.
problem Understanding the structure of compact ECS manifolds.
method Review of basic facts, construction of examples, and proof of a topological structure result.
result Compact rank-one ECS manifolds are bundles over S^1 with fibers being leaves of D⊥.
The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S^3 branched over the closed braid. Every (null homologous) genus one fibered knot in a 3-manifold may be obtained in this way. Using this perspective we answer a question of Morimoto about the number of genus one fibered knots …
Proves global existence and uniqueness of solutions for Einstein-scalar-field equations.
problem Global existence and uniqueness of solutions for specific Einstein-scalar-field equations.
method Proves global existence and uniqueness of classical solutions with small initial data and wake-like decaying null infinity.
result Global existence and uniqueness of solutions for the equations with wake-like decaying null infinity.
This note presents an analytic construction of the optimal unit-norm direction hat(x) = x/|x| that maximizes or minimizes the objective linear expression, B . hat(x), subject to a system of linear constraints of the form [A] . x = 0, where x is an unknown n-dimensional real vector to be determined up to an overall norm…
We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …
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 …
In this study, we define a family of null curves in Minkowski 3-space and called null similar curves. We obtain some properties of these special curves. We show that two null curves are null similar curves if and only if these curves form a null Bertrand pair. Moreover, we obtain that the family of null geodesics and n…
Study transverse metric expansion on null hypersurfaces, proving uniqueness for Killing horizons.
problem Analyzing transverse expansion of metric on null hypersurfaces.
method Covariant approach, general geometric identities, generalized symmetry generators.
result Transverse expansion of spacetime metric uniquely determined at non-degenerate Killing horizons.
The paper explores geometric invariants of null hypersurfaces using Carrollian geometry.
problem Understanding the thermodynamics of black hole solutions.
method Examining various Carrollian geometries and their connections to null hypersurface embeddings.
result A connection with torsion is the most natural object to study Carrollian manifolds.
Estimates bandwidth for CMC initial data sets.
problem Estimating bandwidth for constant mean curvature (CMC) initial data sets.
method Three independent proofs: stability of null expansion, spacetime harmonic function perturbation, Dirac operator.
result Generalized Gromov's band width estimate to CMC initial data sets.
Double descent phenomenon explained in simple terms.
problem Understanding the surprising drop in test error in overparameterized models.
method Informal explanation using linear algebra and probability, visual intuition with polynomial regression, mathematical analysis with ordinary linear regression.
result Three factors create double descent: data undersampling, model size, and parameter count. Ablating any one of these factors prevents double descent.
Identifies null hypersurfaces with constant surface gravity.
problem Understanding null hypersurfaces in spacetimes.
method Analyzes spacetimes satisfying null convergence condition.
result Null hypersurfaces admit null sections with constant surface gravity.
Paper proves existence of ambient manifolds for null hypersurfaces solving Einstein equations.
problem Analyzing transverse expansion of metrics at null hypersurfaces.
method Covariant approach proving existence of ambient manifolds given asymptotic expansion and constraint equations.
result Existence of ambient manifolds solving Einstein equations to infinite order at null hypersurfaces.
Study optimal transport on null hypersurfaces and null energy condition.
problem Optimal transport degeneracy on null hypersurfaces.
method Developed tools to characterize null energy condition using convexity properties of entropy.
result Optimal transport characterization of null energy condition.
In this paper, we define the notion of eikonal helix and eikonal slant helix for null curves in the 4-dimensional Lorentzian manifold M 1 4 and give a characterization for the null curve to be the null eikonal helix. Moreover, we indicate an important relation between the null eikonal helix and null eikonal slant helix…
Study on null helices in semi-Riemannian manifolds with special submanifolds.
problem Investigating geometric properties of null helices on totally umbilical submanifolds in 3D semi-Riemannian manifolds.
method Using the null Frenet frame and degenerate metric condition, equations and invariants characterizing null helices are derived.
result Equations and invariants characterizing null helices on totally umbilical submanifolds in 3D semi-Riemannian manifolds are obtained.