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.
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.
Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.
problem Analyzing singularity formation and asymptotic behavior of null mean curvature flow.
method Rescaling procedure to relate to 2d-Ricci flow, singularity analysis, asymptotic behavior study.
result Ancient solutions to the flow can be understood in terms of 2d-Ricci flow.
Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r-null submanifolds in indefinite Sasakian space forms. 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.
New findings on curvature and null spaces of Laplacians.
problem Relationship between sectional curvature and Laplacian null spaces.
method Analysis of curvature operators and Laplacians on Riemannian manifolds.
result Curvature operator's positivity implies sectional curvature positivity.
In this paper, we investigate the null (light-like) sectional curvatures of Lorentzian warped product manifolds. We derive the formulas for the null sectional curvature of many well-known warped product space-time models such as multiply generalized Robertson-Walker space-times, generalized Kasner spacetimes and standa…
The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the traje…
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.
Proves inextendibility of weak null singularities from curvature blow-up.
problem Inextendibility of weak null singularities in the context of curvature blow-up.
method Introduces a new strategy to infer Cloc0,1-inextendibility from curvature blow-up. result Expected to contribute to the resolution of strong cosmic censorship conjecture.
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
New types of null hypersurfaces found in Sasakian space-forms.
problem Identifying new structures in Sasakian space-forms.
method Defined and analyzed contact screen conformal and umbilic null hypersurfaces.
result Proved these hypersurfaces exist in Sasakian space forms with specific curvature.
We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in E14. 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.
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.
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 nul…
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…
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.
Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. 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 intere…
Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.
Study of null hypersurface foliation in low regularity spacetimes.
problem Control the geometry of null hypersurfaces in low regularity spacetimes.
method Generalised methods of Klainerman and Rodnianski, Alexakis, Shao and Wang.
result Locally uniformly bounded ingoing and outgoing null expansions.
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…
We introduce the notion of k-type slant helix in Minkowski space $\e_1^4$. For partially null and pseudo null curves in $\e_1^4$, we express some characterizations in terms of their curvature and torsion functions.
Study of null hypersurfaces in semi-Riemannian manifolds with applications.
problem Characterize null hypersurfaces in semi-Riemannian manifolds with varying sectional curvature.
method Introduce screen quasi-conformal hypersurfaces and extend results to non-zero sectional curvature.
result Classification results for umbilical, isoparametric, and Einstein null hypersurfaces in Lorentzian space forms.
The paper explores equi-affine curvatures in pseudo-Riemannian manifolds.
problem Understanding equi-affine curvatures in pseudo-Riemannian manifolds.
method Using Cartan frames and Frenet frames, the paper describes equi-affine curvatures and their relation to Frenet curvatures.
result The constancy of Frenet curvatures does not guarantee the constancy of equi-affine curvatures, and vice versa.
Extends Penrose's method to null shells with pressure and energy flux.
problem Constructing null thin shells with arbitrary gravitational/matter content.
method Derive locally Lipschitz metric and coordinate transformation.
result Example of null shell with non-trivial energy density, flux, and pressure in Minkowski space.
Study on null hypersurfaces with constant angle in Lorentzian manifolds.
problem Understanding constant angle null hypersurfaces in Lorentzian manifolds.
method Introduced constant angle null hypersurfaces, analyzed with respect to a given ambient vector field, and provided classification results.
result Null hypersurfaces have a canonical principal direction when the vector field is closed and conformal.
Minimal surfaces in Heisenberg group have null curves and lines.
problem Characterizing timelike minimal surfaces in the Heisenberg group.
method Characterization through null curves and lines with prescribed curvatures.
result Minimal surfaces are defined by the multiplication of null curves and affine null lines.
The paper studies special null hypersurfaces in spacetimes.
problem Characterizing null screen isoparametric hypersurfaces in Lorentzian space forms.
method Developed screen isoparametric hypersurface concept for null hypersurfaces of Robertson-Walker spacetimes, derived Cartan identities, and provided local characterizations.
result Derived Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provided a local characterization.
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 the…
The paper extends Hawking--Page solutions to various spacetimes with singularities.
problem Understanding the extensions of Hawking--Page solutions with different types of singularities.
method Kaluza--Klein reduction and Christodoulou's methods.
result Extensions of Lorentzian Hawking--Page solutions with null, spacelike singularities, and Cauchy horizons of Taub--NUT type are proven.
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface ∂Ω0, we show that there exists a weak solution to the null mean curvatu…
We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold M and the Jacobi operators with respect to particular spacelike unit vectors on M. We study the number of the eigenvalues of such operators in a φ-null Osserman Lorentzi…
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.
Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…
New perspective on Ricci flow on spheres using Minkowski spacetime.
problem Classifying singularity models for null mean curvature flow in Minkowski spacetime.
method Equivalence of 2d-Ricci flow and null mean curvature flow on lightcones.
result Classification of singularity models for null mean curvature flow.
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.
Paper classifies timelike Thomsen surfaces and reveals their geometric properties.
problem Classifying timelike Thomsen surfaces and understanding their geometric properties.
method Use of Lorentz conformal and null coordinates, representation theorems, and geometric invariants.
result Revealed the relationship between timelike Thomsen surfaces and timelike minimal surfaces with planar curvature lines.
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.
Constructs surfaces with constant mean curvature in Schwarzschild spacetime near null infinity.
problem Creating surfaces with constant mean curvature in the Schwarzschild spacetime near null infinity.
method Constructs spacelike surfaces as graphs of functions u(y, r) with specified asymptotic behavior.
result The constructed surfaces intersect future null infinity with a cut given by a function f.
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.
Study of null φ-slant curves in specific 3D manifolds.
problem Characterizing null φ-slant curves in 3D normal almost contact B-metric manifolds.
method Analyzing the geometric properties and Frenet frames of φ-slant null curves.
result Existence of a unique Frenet frame for non-geodesic φ-slant null curves.
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. Null geodesics spaces can be embedded into globally hyperbolic spacetimes.
problem Obstructing conformal embeddings of causally simple spacetimes.
method Analyzing null geodesics and conformal embeddings.
result Causally simple spacetimes can be non-conformally embeddable into globally hyperbolic ones.
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.
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…