arXiv research

Study of mean curvature flow on null hypersurfaces leading to MOTS.

The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.

The paper studies null hypersurfaces in 4-manifolds with a specific metric structure.

Study on convex surfaces in Minkowski space, proving completeness and incompleteness conditions.

New findings on how conformal rescalings affect spacetime metrics.

Study optimal transport on null hypersurfaces and null energy condition.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

Note shows equivalence of recent NEC reformulation to classical NEC for $C^2$-metrics.

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…

The study examines null curves in specific geometric manifolds and their properties.

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 $\mathcal{S}$-manifolds. We prove that a Lorentz $\mathcal{S}$-manifold with constant…

Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.

Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.

Study on null helices in semi-Riemannian manifolds with special submanifolds.

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…

Study of null φ-slant curves in specific 3D manifolds.

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…

Extends results on marginally outer trapped surfaces to general null expansion.

New method to find surfaces in null cones with constant curvature near black hole indicators.

The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.

Proves the Kundt conjecture in arbitrary dimensions, confirming its validity.

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…

In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and s…

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

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…

We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide severa…

We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces satisfying the null convergence condition.

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…

In this paper, we investigate the tangent indicatrix of the curve C with constant curvature. Tangent indicatrix of the curve C is characterized with det(C^(3),C^(4),C^(5))=0 in Minkowski 3-space E13. Moreover, we study null slant helices using the determinant approach and give the following characterization: A curve C …

New findings on curvature and null spaces of Laplacians.

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 paper develops theory for holomorphic null curves in SL2(C).

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…

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riema…

Proves inextendibility of weak null singularities from curvature blow-up.

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…

We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.

Study null curves and their motion in 3D flat space-time, leading to integrable hierarchies.

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

Unified approach to various energy conditions in spacetime geometry.

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…

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a `past null focusing' condition. Such a condit…

Study on null hypersurfaces in complex contact manifolds.

Identifies null hypersurfaces with constant surface gravity.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.