Paper extends previous result on hypersurfaces with degenerate light-like points.
problem Characterizing hypersurfaces with degenerate light-like points in Lorentzian manifolds.
method Analyzes C3-differentiable hypersurfaces, extending previous C4-differentiability result. result Same conclusion holds for C3-differentiable hypersurfaces as for C4-differentiable ones. Consider a constant mean curvature immersion F:U(⊂Rn)→M into an arbitrary Lorentzian (n+1)-manifold M. A point o∈U is called a light-like point if the first fundamental form ds2 of F degenerates at o. We denote by BF the determinant function of the symmetric matrix associate…
With several concrete examples of zero mean curvature surfaces in R13 containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first …
Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.
problem Proving entire zero mean curvature graphs are hyperplanes in Lorentz-Minkowski space.
method Using line theorems at degenerate light-like points to generalize Bernstein theorem.
result Entire zero mean curvature graphs in Lorentz-Minkowski space are hyperplanes if they only contain space-like or light-like points.
The paper studies sections of time-like twistor spaces with specific covariant derivatives.
problem Sections of time-like twistor spaces with light-like or zero covariant derivatives.
method Analyzes conformal Gauss maps of time-like minimal surfaces and properties of almost paracomplex structures.
result Sections of time-like twistor spaces have light-like or zero covariant derivatives.
In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …
In this paper, we are concerned with light-like extremal surfaces in curved spacetimes. It is interesting to find that under a diffeomorphic transformation of variables, the light-like extremal surfaces can be described by a system of nonlinear geodesic equations. Particularly, we investigate the light-like extremal su…
Proves energy quantization for surfaces with bounded index.
problem Energy quantization for Willmore surfaces with bounded index.
method Translated the question to the conformal Gauss map's perspective and showed convergence in specific regions.
result Conformal Gauss map converges to a light-like geodesic in De Sitter space in neck or collar regions.
It is well-known that a torsion-free linear connection on a light-like manifold (M,g) compatible with the degenerate metric g exists if and only if Rad(TM) is a Killing distribution. In case of existence, there is an infinitude of connections with none distinguished. We propose a method to single out connections …
Study classifies translating solitons in Minkowski 3-space, revealing singularities.
problem Classifying translating solitons in Minkowski 3-space.
method Introduced the concept of translating solitons on a light-like direction, classified them into graphical and invariant families.
result All time-like examples are incomplete and some have singularities.
It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…
We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane.
Paper finds space-like maximal surfaces with entire null lines in 3D space-time.
problem Existence of space-like maximal surfaces containing entire null lines.
method Analyzes surfaces in Lorentz-Minkowski 3-space, proving existence and properties.
result Embedded space-like maximal graphs containing entire null lines exist.
The paper classifies submanifolds in a specific Lorentz-Minkowski space.
problem Characterizing submanifolds in a Lorentz-Minkowski space.
method Constructing a global frame field and analyzing extrinsic invariants.
result Local classification theorems for specific submanifold classes.
We consider Lorentzian manifolds with parallel light-like vector field V. Being parallel and light-like, the orthogonal complement of V induces a codimension one foliation. Assuming compactness of the leaves and non-negative Ricci curvature on the leaves it is known that the first Betti number is bounded by the dimensi…
In L^3, cuspidal edges can have bounded mean curvature under specific conditions.
problem Understanding cuspidal edges with bounded mean curvature in Lorentz-Minkowski 3-space.
method Investigated cuspidal edges and generalized cuspidal edges, analyzing their singular points and principal curvatures.
result Cuspidal edges with bounded mean curvature in L^3 occur only when the singular set is a light-like curve.
We investigate Kaluza-Klein metrics with a recurrent light-like vector field over a pseudo-Riemannian manifold.
We consider variation of energy of the light-like particle in Riemann space-time, find lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is shown to not lea…
In this paper, we characterize and classify all surfaces endowed with canonical principal direction relative to a space-like and light-like, constant direction in Minkowski 3-spaces.
Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.
problem Proves a new Bernstein-type theorem for space-like zero mean curvature graphs.
method Uses fluid mechanical duality between minimal surfaces and maximal surfaces.
result Shows that a zero mean curvature graph with only space-like and light-like points is a plane.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.
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…
Totally geodesic null hypersurfaces found in Lorentzian manifolds.
problem Characterizing null hypersurfaces in Lorentzian manifolds.
method Analyzing light-like geodesically complete Lorentzian manifolds with null energy condition.
result Null hypersurfaces are totally geodesic.
Proves compatibility of light cones and projective structures.
problem Clarifying different concepts of compatibility between conformal and projective structures.
method Analyzes compatibility criteria introduced by Ehlers-Pirani-Schild and Trautman-Scholz.
result Proves that the compatibility criterion introduced by Ehlers-Pirani-Schild is correct.
Study on surfaces in neutral space forms with zero mean curvature.
problem Characterizing surfaces with zero mean curvature in neutral space forms.
method Analyzing curvature and normal connection properties of time-like conformal immersions.
result Conditions for surfaces with zero mean curvature in neutral space forms.
A pp-wave is a Lorentzian manifold with a parallel light-like vector field satisfying a certain curvature condition. We introduce generalisations of pp-waves, on one hand by allowing the vector field to be recurrent and on the other hand by weakening the curvature condition. These generalisations are related to the scr…
Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure.…
The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.
problem Understanding null hypersurfaces in Lorentz-Minkowski space.
method Analyzing L-complete null hypersurfaces as wave fronts in Euclidean space. result Most null wave fronts can be realized as restrictions of certain L-complete null wave fronts. Study on extremals in sub-Lorentzian geometry defined by antinorm.
problem Characterizing extremals in sub-Lorentzian structures.
method Deriving Hamiltonian system and conditions for extremal trajectories.
result Conditions for normal extremal trajectories and properties of abnormal extremals.
The paper classifies time-like surfaces in a static space-time.
problem Classifying time-like surfaces in a static space-time.
method Constructing a pseudo-orthonormal frame field and analyzing invariants.
result Complete classification theorem for class~A surfaces. We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like co…
Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.
Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.
problem Projective compactness for torsion-free linear connections on a manifold.
method Introduce and study a weakening of projective compactness for torsion-free linear connections on a manifold.
result Induces projective structure on the boundary and relates to asymptotic forms in GR.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
Study of generalized Bishop frames on time-like curves in 4D Lorentz space.
problem Characterize frames for time-like curves in 4D Lorentz space.
method Introduced and studied generalized Bishop frames for regular time-like curves in 4D Lorentz space.
result Hierarchy of frames exists for time-like curves in 4D Lorentz space, similar to Euclidean case.
Novel contact metric structures lead to supergravity solutions.
problem Developing new contact metric structures for supergravity.
method Introducing and studying εη-Einstein structures. result Constructed families of six-dimensional supergravity solutions.
New manifolds found without interior conjugate points.
problem Existence of interior conjugate points in hyperbolic manifolds.
method Construction of non-trapping asymptotically hyperbolic manifolds.
result Found manifolds without interior conjugate points.
Estimator calculates surface curvature from point cloud samples.
problem Accurately estimating curvature from limited point cloud data.
method Algorithm using probability distribution and nearby points control.
result Controlled number of points ensures accurate curvature estimation.
New tools for constructing fixed point sets in digital topology.
problem Constructing fixed point sets in digital topology.
method Defining excludable points and articulation points, and showing their exclusion from freezing sets.
result Excludable points and articulation points can be excluded from all freezing sets.
This paper analyzes saddle points and minimax points in non-convex smooth games.
problem Understanding local optimal points in non-convex smooth games.
method Comprehensive analysis of local minimax points, including their optimality conditions and stability.
result Local saddle points are uniformly local minimax points under mild continuity assumptions.
While Multiple Instance (MI) data are point patterns -- sets or multi-sets of unordered points -- appropriate statistical point pattern models have not been used in MI learning. This article proposes a framework for model-based MI learning using point process theory. Likelihood functions for point pattern data derived …
Study on connection points on double regular polygons, providing coordinates and proving non-connection points.
problem Identifying connection points on double regular polygons.
method Examined coordinates in trace field, provided constructive proof for prime n. result For n=7, conjectured all remaining points are connection points; for n≥7 prime, provided explicit separatrix. New families of translation surfaces with multiple oblivious points discovered.
problem Identifying points on translation surfaces without nearby closed geodesics.
method Constructing new families of translation surfaces and proving existence in higher genera.
result Translation surfaces in every genus ≥3 have at least one oblivious point.
This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
A new model for point processes without intensity function trade-offs.
problem Inefficiency and trade-offs in existing point process models.
method Point Set Diffusion, a diffusion-based latent variable model.
result Achieves state-of-the-art performance in point process generation.
PINNACLE optimizes point selection for PINNs, improving accuracy.
problem Challenges in selecting points for training Physics-Informed Neural Networks (PINNs).
method Introduces PINNACLE, an algorithm that jointly optimizes collocation and experimental points selection, adjusting point proportions dynamically.
result PINNACLE outperforms existing methods in forward, inverse, and transfer learning problems.
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.