Generalizes Fermat's principle for wave propagation in cone structures.
problem Wave propagation in complex media with discontinuities and anisotropy.
method Generalizes Fermat's principle to smooth interfaces separating two cone structures representing wave propagation in various media.
result Conditions for critical points of arrival time functional, generalizing Snell's law and reflection.
Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.
problem Analyzing mixed types of curves with singular points in Lorentz-Minkowski 3-space.
method Using lightcone frame to consider Bertrand types for lightcone framed curves.
result Existence conditions of Bertrand lightcone framed curves in all cases.
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.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
A new model uses Lorentz-Finsler geometry to predict wave propagation.
problem Modeling wave propagation in anisotropic and rheonomic media.
method Identifying wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, solving ODE systems.
result Wave trajectories can be easily computed in real time.
Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.
problem Investigate differential geometric properties of lightcone framed surfaces.
method Introduced modified frame to study the properties of lightcone framed surfaces.
result Showed behavior of Gaussian and mean curvatures at lightlike and singular points.
We define the notions of St1×Ss1-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of s…
Past lightcones of certain points in globally hyperbolic spacetimes determine the entire spacetime.
problem Determining the entire spacetime from the past lightcone of a point.
method Analyzing properties of globally hyperbolic spacetimes and using null lines and observer horizons.
result Past lightcones of certain points in globally hyperbolic spacetimes determine the entire spacetime (up to isometry).
New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.
problem Understanding the geometry of weighted Lorentz-Finsler manifolds.
method Developed a splitting theorem using weighted Berwald spacetimes and Busemann functions.
result Weighted Lorentz-Finsler manifolds with certain properties split into simpler isometric translations.
We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the e…
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.
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
Unified geometry for relativity and beyond.
problem Unified geometric framework for relativity and beyond.
method Unified Lorentz-Finsler geometry.
result Unified framework for relativity and beyond.
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.
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. On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must b…
Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.
problem Creating foliations of lightcones with specific geometric properties.
method Employing a geometric flow inspired by Huisken-Yau's approach for Riemannian settings.
result Initial data converges exponentially to an STCMC surface under area preserving null mean curvature flow.
The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel…
Paper proves uniqueness of specific spacetime surfaces in a lightcone.
problem Proving uniqueness of surfaces of constant spacetime mean curvature in a lightcone.
method Used a fairly generic notion of asymptotic flatness to prove uniqueness.
result Unique foliation by surfaces of constant spacetime mean curvature exists under weaker assumptions.
We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern-Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced.
Introduces a variational framework for indefinite Lagrangians with specific symmetries.
problem Handling indefinite Lagrangians with complex symmetries.
method Develops a variational setting for an indefinite Lagrangian with a specific Noether charge.
result Validates the existence of a variational setting for a broad class of Lagrangians.
Revisits Finsler spacetimes from inertial observer perspective.
problem Physical foundations of relativistic spacetimes.
method Inertial observers and double linear approximation.
result Finsler spacetimes are defined by dropping the second linearization.
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 shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
Study on stability of surfaces in null cones under area-preserving variations.
problem Investigating stability of spacelike cross sections of null cones.
method Area-preserving variations, Hawking energy analysis, spherical cross sections.
result Only round spheres are stable cross sections of the standard Minkowski lightcone.
Study of quasilocal energy in higher dimensions, focusing on small sphere limits.
problem Understanding quasilocal energy in higher dimensions and its small sphere limits.
method Generalized quasilocal energy definitions, evaluated along lightcone cuts, and compared with known energies.
result The small sphere limits of quasilocal energy in higher dimensions are not proportional to the Bel-Robinson superenergy, challenging its role as gravitational energy.
Any 2-dim Riemannian manifold with spherical topology can be embedded isometrically into a lightcone of the Minkowski spacetime. We apply this fact to give a proof of the Kazdan-Warner identity.
We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.
Wave propagation framework using cone structures and observers' vector fields.
problem Describing classic wave propagation in anisotropic media.
method Introduces a cone structure C and an observers' vector field ∂t to describe wave propagation. result Reduces the PDE for wavefronts to ODE for cone geodesics of C. A new model predicts wildfire spread with wind and slope effects.
problem Predicting wildfire spread with wind and slope effects.
method Geometric model based on Lorentz-Finsler framework, considering wind and slope.
result Infinitesimal wavefronts are no longer restricted to be elliptical, allowing for more accurate predictions.
Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For …
A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field η. Several sufficient assumptions on such a surface with non-degenerate η-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…
We prove a rigidity of the lightcone in Minkowski space. It is essentially the unique space endowed with a degenerate Riemannian metric, of lightlike type, and supporting an isometric non-proper action of a semi-simple group.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
In a recent work I showed that the family of smooth steep time functions can be used to recover the order, the topology and the (Lorentz-Finsler) distance of spacetime. In this work I present the main ideas entering the proof of the (smooth) distance formula, particularly the product trick which converts metric stateme…
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to (1+n)-dimensional Lorentz--Finsler manifolds. result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
To study spacelike surfaces in the Lorentz-Minkowski space R14, we construct a pair of maps whose values are in the lightcone, called lr±-Gauss maps. We can use these maps to study umbilical spacelike surfaces and find parametrizations of spacelike surfaces of revolution of hyperbolic and …
The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal tran…
We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…
In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points …
We study Finsler spacetimes and Killing vector fields taking care of the fact that the generalized metric tensor associated to the Lorentz-Finsler function L is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vect…
The paper characterizes global hyperbolicity in Lorentzian manifolds without relying on manifold topology.
problem Characterizing global hyperbolicity in smooth Lorentzian manifolds without assuming manifold topology.
method Two formulations of global hyperbolicity: one using chronological diamonds and the other using properties of the Lorentzian distance function.
result The second formulation is equivalent to the definition of `Lorentzian metric space' and introduces the concept of d-reflectivity. 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.
We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These…
Gauging procedure constructs lagrangians for carrollian gravity.
problem Constructing lagrangians for carrollian gravity.
method Gauging procedure applied to Klein pairs corresponding to homogeneous spaces.
result Generalizes first-order lagrangians for four-dimensional maximally symmetric carrollian spaces.
A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics L is carried out. As a link between both notions, cone triples (Ω,T,F), where Ω (resp. T) is a 1-form (resp. vector field) with Ω(T)≡1 and F, a Finsler metric on ker(Ω), are introduced. Explicit descriptions o…
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
Study of spacelike discs in Minkowski cones, proving self-similar expansion.
problem Mean curvature flow of spacelike discs in Minkowski cones.
method Analysis of parabolic boundary value problem for self-similar solutions.
result Existence of solutions rescaling to self-similarly expanding solutions.