arXiv research

Study of Moncrief lines' behavior in curved space-times.

Proves existence of Killing fields in smooth spacetimes with compact Cauchy horizons.

We obtain basic estimates for a Monge-Ampère equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmü…

We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to …

In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and…

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates to…

We complement a recent work on the stability of fixed points of the CMC-Einstein-$Λ$ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we…

We study the geometry of the foliation by constant Gaussian curvature surfaces $(Σ_k)_k$ of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizati…

New method calculates volume-renormalized mass from Hamiltonian perspective.

New insights into black hole horizons from asymptotic expansions.

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumpt…

The study classifies compact Cauchy horizons in vacuum spacetimes.

New proof shows perturbed non-compact Einstein spaces attract to unique global solution.

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, r…

The paper describes the geometric properties of line congruences' singularities.

Study restricts line arrangements with odd points using topological arguments.

A special group of transformations of the real line cannot act effectively on it.

Study Blaschke's asymptotic lines on surfaces in 3D space.

Co-PLNet combines point and line predictions to improve wireframe parsing accuracy and efficiency.

Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.

New Calabi-Yau metrics with conical singularities are created near complex lines.

For a pair of points in a smooth closed convex planar curve $γ$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Sy…

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is…

The paper examines asymptotic lines of plane fields in 3D space.

The paper explores reflection principles for lightlike line segments on maximal surfaces.

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

The paper explores graphons of line graphs from sparse finite graphs.

New method describes entanglement of straight lines in 3D space.

Shows CM line bundles are ample on K-stable varieties.

Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.

Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial,…

The paper finds two types of metric lines in curve spaces.

We define a pseudo-inverse for line graphs using linear integer programming.

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …

Geometric quantization extended to big line bundles.

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…

New BDEs reveal singular surfaces from line congruences.

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element is a smooth surface containing $E$ and no other line. Afterwards we prove that …

Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.

Study conic-line arrangements of degree 7, finding their topology and components.

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…

The paper classifies singularities of line congruences in 4D space.

Study three discrete envelope types of polygon bisection lines.

The Samuelson condition is not satisfied by tangent lines of quadratic curves.

Study curvature lines of a vector field on surfaces.

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.

New framework for analyzing line fields on surfaces, proving stability under specific conditions.