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.
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.
The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.
problem Characterizing the null energy condition in Lorentzian manifolds.
method Characterization via convexity of the relative entropy along displacement interpolations on null hypersurfaces.
result The null energy condition is characterized in terms of convexity of the relative entropy.
Let M1n+1 be a light-like geodesically complete Lorentzian (n+1)-manifold satisfying the null energy condition. We show that null hypersurfaces properly immersed in M1n+1 are totally geodesic.
The paper studies marginally trapped submanifolds in Lorentzian manifolds under null energy condition.
problem Understanding marginally trapped submanifolds in Lorentzian manifolds.
method Analyzes properties of marginally trapped submanifolds in a Lorentzian manifold satisfying the null energy condition.
result Marginally trapped submanifolds have locally volume-maximizing properties in certain null hypersurfaces.
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…
Synthetic framework for null hypersurfaces in non-smooth spacetimes.
problem Analyzing null hypersurfaces in non-smooth spacetimes.
method Develops synthetic null hypersurfaces using optimal transport and Lorentzian geometry.
result Synthetic null energy condition stabilizes under convergence and applies to low-regularity spacetimes.
In this paper, we study inextensible flows of partially null and pseudo null curves in E_1^4. We give neccessary and sufficent conditions for inextensible flows of partially null and pseudo null curves in E_1^4
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, …
The main result we give in this brief note relates, under suitable hypotheses, the φ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.
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.
Proves compact Cauchy horizons have constant surface gravity under null energy condition.
problem Proving compact Cauchy horizons have constant surface gravity.
method Combines ergodic theory, Hodge theory, and Riemannian flow theory.
result Compact Cauchy horizons admit a smooth lightlike tangent vector field of constant surface gravity.
We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its pseudodistancfe is nontrivial. If its pseudodistance is nondegenerate, all of its null geodesi…
The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.
problem Determining when a submanifold lies on a shear-free null hypersurface under integral curvature conditions.
method Using Minkowski formulas with arbitrary weight to derive rigidity results for submanifolds with weaker integral curvature conditions.
result A necessary and sufficient condition for a submanifold to lie in a shear-free null hypersurface is given by a mean curvature integral inequality.
Study null energy condition impacts on special hypersurfaces in static spacetimes.
problem Effects of null energy condition on totally umbilic hypersurfaces.
method Characterization of embedded surfaces and photon surfaces using Alexandrov Theorem and other methods.
result Full characterization of embedded surfaces with constant spacetime mean curvature.
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…
A condition of Osserman type, called φ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz S-manifolds. We prove that a Lorentz S-manifold with constant…
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.
New findings on how conformal rescalings affect spacetime metrics.
problem Understanding how conformal rescalings impact spacetime metrics.
method Analyzing the null curvature condition and causal structure.
result Proving constraints on conformal rescalings in vacuum and non-vacuum spacetimes.
Study on convex surfaces in Minkowski space, proving completeness and incompleteness conditions.
problem Characterizing isometric embeddings of hyperbolic plane in Minkowski 3-space.
method Analysis of null support function and curvature conditions.
result Conditions for completeness and incompleteness of convex surfaces.
The paper studies null hypersurfaces in 4-manifolds with a specific metric structure.
problem Characterizing geometric properties of null hypersurfaces in 4-manifolds.
method Analyzes hypersurfaces null with respect to a neutral metric derived from a Riemannian Einstein metric and an almost paracomplex structure.
result Shows that totally geodesic null hypersurfaces imply Ricci-flatness of the ambient Einstein metric and provides necessary conditions for other types of null hypersurfaces.
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.
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.
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.
In [21], the authors initiated the study of quasi generalized CR (QGCR)-null submanifolds. In this paper, attention is drawn to some distributions on ascreen QGCR-null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finall…
Extends results on marginally outer trapped surfaces to general null expansion.
problem Analyzing geometry and topology of expanding horizons.
method Introduces g-stability and proves conditions for positive Yamabe type and scalar curvature. result Initial data sets with compact boundary of positive null expansion have positive mass.
On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions…
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.
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
Simplified proof of cosmic singularity theorem using new mathematical techniques.
problem Proving cosmic singularity in expanding spacetimes with positive cosmological constant.
method Unified approach using the positive resolution of the virtual positive first Betti number conjecture.
result The theorem holds without the need for a spherical Cauchy surface.
Null-homotopic knots in certain 3-manifolds are uniquely identified by their complements.
problem Identifying null-homotopic knots in specific 3-manifolds.
method Instanton Floer homology and SU(2)-representation varieties.
result Null-homotopic knots are determined by their complements in certain 3-manifolds.
Note shows equivalence of recent NEC reformulation to classical NEC for C2-metrics.
problem Consistency of null energy condition in Lorentzian length spaces.
method Shows equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. result Equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. 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.
In this paper we study null Bertrand curves in R14 under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in R14 is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R…
Synthetic proof of Gannon-Lee theorem for spacetimes.
problem Proving incompleteness in globally hyperbolic spacetimes.
method Synthetic null energy condition and synthetically asymptotically regular trappedness condition.
result Generalized classical incompleteness theorem to weighted spacetimes.
Study of Bondi-Sachs formalism for massless scalar field with zero cosmological constant.
problem Analyzing the Bondi-Sachs formalism for Einstein's massless scalar field equations.
method Asymptotic expansions and peeling property for Bondi-Sachs metrics and scalar fields.
result Positivity of Bondi energy-momentum under specific conditions.
Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
problem Characterize submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
method Analyze light cones, lightlike cylinders, and null cones in specific spacetimes; provide conditions for conformal diffeomorphism.
result Conditions guaranteeing conformal diffeomorphism to hyperbolic space, round cylinder, and sphere.
Null Kähler metrics are characterized by Painlevé I or II ODEs.
problem Characterizing null-Kähler metrics in four dimensions.
method Cohomogeneity-one anti-self-dual null-Kähler metrics, twistor methods, Painlevé I and II ODEs.
result Cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterized by solutions to Painlevé I or Painlevé II ODEs.
Null-Calibrated Conformal Selection via Target-Membership Scores
problem Identifying test candidates whose unknown responses fall in a target region while controlling the false discovery rate
method Membership-score-based conformal selection
result Finite-sample valid null p-values
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.
The study examines null curves in specific geometric manifolds and their properties.
problem Characterizing null curves in Sasaki-like almost contact B-metric manifolds.
method Expressed Frenet frames and curvatures, proved curvature constancy conditions, and found necessary conditions for generalized helices and null cubic.
result Curvatures of specific null curves are constant if a function on the manifold is constant.
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 proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.
problem Analyzing spacelike CMC surfaces near future null-infinity in Schwarzschild spacetime.
method Proves asymptotic hyperbolicity, derives boundary data expressions, and shows compatibility conditions.
result Compatibility conditions and asymptotic behaviors of spacelike CMC surfaces near future null-infinity.
In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jo…
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.
The paper studies convergence of cosmological spacetimes using null distance.
problem Convergence of cosmological spacetimes with compact slices.
method Using null distance and Gromov-Hausdorff convergence, the paper establishes convergence results for spacetimes with mild extension properties.
result Uniform convergence of null distances and Gromov-Hausdorff convergence for monotone sequences of spacetimes.
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 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 spac…