The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.
problem Extracting scientific understanding from particle-physics experiments with high precision and efficiency.
method L-GATr, a geometric algebra Transformer, representing data in 4D space-time and being equivariant under Lorentz transformations.
result L-GATr achieves performance comparable to or better than domain-specific baselines on regression, classification, and generative tasks.
Study examines how Lorentz transformations affect foliations in spacetime.
problem Understanding foliations and their relation to observers under Lorentz transformations.
method Exploration of kinematic characteristics of timelike congruences (observers) and their relation to foliations.
result Local Lorentz transformations impact the existence of transversal foliations.
While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…
We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of…
The paper proves a stability result for translating space-like graphs in Lorentz manifolds.
problem Investigating stability of translating space-like graphs in Lorentz manifolds.
method Analyzing space-like graphs over a domain in Lorentz manifold with a specific metric and proving stability under conformal transformation.
result An interesting stability result for translating space-like graphs in MnimesR is proven. LLoCa makes any network Lorentz-equivariant, achieving high accuracy and efficiency.
problem Limitations of specialized layers in Lorentz-equivariant neural networks.
method LLoCa framework using local reference frames and geometric message passing.
result Models achieve competitive and state-of-the-art accuracy on particle physics tasks.
The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.
problem Classifying curves with constant curvature in dual affine and Lorentz-Minkowski planes.
method Investigation of invariants under equiaffine transformations and explicit equations for curves with constant curvature.
result Curves with constant curvature in dual affine and Lorentz-Minkowski planes are classified.
This is the first part of a series on non-compact groups acting isometrically on compact Lorentz manifolds. This subject was recently investigated by many authors. In the present part we investigate the dynamics of affine, and especially Lorentz transformations. In particular we show how this is related to geodesic fol…
The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.
problem Investigating the properties and singularities of helicoidal surfaces in Lorentz-Minkowski space.
method Defining and analyzing two types of helicoidal surfaces, using diffeomorphic transformations and criteria for cusps and cuspidal edges.
result Identification theorems for the singular types of both 1-type and 2-type helicoidal surfaces.
Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.
problem Investigating minimal surfaces in Lorentz-Minkowski space.
method Complex and paracomplex analysis, Möbius-type transformations, pseudo-isometries.
result Unified approach to both spacelike and timelike minimal surfaces.
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in R24. method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
Proves estimate similar to De Lellis-Müller on Minkowski lightcone.
problem Estimating spacelike cross sections of the Minkowski lightcone.
method Geometric scaling invariant estimate, using singularity models and almost-Schur lemma.
result Spacelike cross sections are W2,2-close to a round surface. Equivariant neural network simplifies particle physics models.
problem Complexity and interpretability in particle physics classification.
method Lorentz group equivariant neural network architecture.
result Simplified, interpretable models with fewer parameters.
Unified study of harmonic maps between pseudo-Riemannian surfaces.
problem Classifying harmonic maps between pseudo-Riemannian surfaces.
method Unified formalism and Bäcklund transformation.
result Unified solutions to harmonic map equations and corresponding maps.
We consider the region of closed timelike curves (CTC's) in three-dimensional flat Lorentz spacetimes. The interest in this global geometrical feature goes beyond the purely mathematical. Such spacetimes may be considered lower-dimensional toy models of sourceless Einstein gravity or cosmology. In particular, our inter…
Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.
Paper proves isoperimetric inequality for Minkowski spacetime.
problem Maximizing volume of domain of dependence for finite lightcone.
method Lorentz polarisation to study variational problem.
result Maximal volume achieved by spacelike hyperplane truncated finite lightcone.
Twistor correspondences for R-invariant indefinite self-dual conformal structures on R^4 are established explicitly. These correspondences are written down by using a natural integral transform from functions on a two dimensional cylinder to functions on the flat Lorentz space R^{1,2} which is related to the wave equat…
Theory of packing diabolic domains in liquid crystals.
problem Understanding the packing of diabolic domains in liquid crystals.
method Lorentz transformations and geometric analysis.
result Diabolic domains can lower the elastic energy of the system.
Sharp Sobolev inequality on circle proven with Lorentz invariance.
problem Proving a sharp Sobolev inequality on the circle.
method Rearrangement inequality, variational method, Lorentz transformation.
result Sharp inequality and global minimizer on circle.
New mappings in Minkowski spacetime classified under mild conditions.
problem Characterizing mappings in Minkowski spacetime.
method Analyzing mappings under minimal assumptions.
result Mappings fall into three categories based on specific conditions.
We simplify supergravity in 10D using geometric insights.
problem Formulating supergravity in 10D without Lorentz degrees of freedom.
method Using generalised geometry, we describe the fibred structure of field space.
result Our action satisfies the classical master equation without Lorentz terms.
Study shows compact Lorentz manifolds can't have closed geodesics.
problem Existence of closed geodesics in compact Lorentz manifolds.
method Constructed compact Lorentz manifolds without closed geodesics.
result Compact Lorentz manifolds can't have closed geodesics.
Paper finds isometric timelike minimal surfaces with unique properties.
problem Rigidity of isometric timelike minimal surfaces in Lorentz-Minkowski space.
method Analyzes symmetries and deformations of timelike minimal surfaces.
result Existence of isometric timelike minimal surfaces not congruent to associated family.
This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; …
The study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.
problem Characterizing Lorentz surfaces in R13. method Introduces canonical isotropic coordinates and a natural equation for the surfaces.
result Proves a Bonnet-type theorem for Lorentz surfaces of general type.
Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.
problem Geodesic completeness of compact locally symmetric Lorentz manifolds.
method Proof in all remaining cases using completeness result.
result All compact, locally symmetric Lorentz manifolds are geodesically complete.
A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.
Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parame…
The three-dimensional Heisenberg group H3 has three left-invariant Lorentz metrics g1, g2 and g3. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking non-gradient Ricci soliton. Likew…
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
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…
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
Paper presents a novel hyperbolic neural network for efficient data representation.
problem Efficient representation of hierarchical data in hyperbolic space.
method Develops a fully hyperbolic neural network using projections and equivariant embeddings.
result Proves the proposed embedding is isometric and equivariant under Lorentz transformations.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper classifies Willmore 2-spheres in Sn.
problem Classifying Willmore 2-spheres in Sn. method Construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map.
result Any Willmore 2-sphere is either totally isotropic or strictly k-isotropic. The paper examines Lorentz Ricci solitons on specific Lie groups.
problem Identifying Lorentz Ricci solitons on 4D non-Abelian nilpotent Lie groups.
method Investigates left-invariant Lorentz metrics on H3imesR and G4. result Only specific metrics on H3imesR and G4 satisfy the Ricci Soliton equation. Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.
problem Characterizing maximal surfaces in Lorentz-Minkowski space.
method Three proofs showing correspondence to minimal surfaces in Euclidean space.
result Conjugate surface of a maximal graph over a convex domain is also a graph.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
We study pseudo-Riemanniasn manifolds (M,g) with transitive group of conformal transformation which is essential, i.e. does not preserves any metric conformal to g. All such manifolds of Lorentz signature with non exact isotropy representation of the stability subalgebra are described. A construction of essential c…
Lecture notes from the mini-course "Topics in Lorentz Geometry" taught at the University of São Paulo, in March/2019. The text has three parts: (i) an overall view of linear algebra in the pseudo-Euclidean space Rνn, with focus on Lorentz-Minkowski space and its role in Physics; (ii) a version of the Funda…
We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature ϰ satisfy the inequality K2−ϰ2>0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …
We show a geometric rigidity of isometric actions of non compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.