The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for C1,1 metrics. Optimizes transport in Finsler spacetimes with lower Ricci bounds.
problem Optimizing transport in Finsler spacetimes with lower Ricci bounds.
method Using optimal transport and weighted Ricci curvature bounds.
result Proves timelike curvature-dimension condition for Finsler spacetimes.
Study timelike Ricci curvature bounds via optimal transport with Orlicz-type costs.
problem Characterize timelike Ricci curvature bounds.
method Optimal transport with Orlicz-type costs, convexity of relative entropy.
result Characterize timelike Ricci curvature lower bounds via convexity of relative entropy.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
New curvature measure for causal sets derived from optimal transport.
problem Capturing Ricci curvature in causal sets.
method Using Lorentzian optimal transport, novel curvature defined along maximal chains.
result Recovery of timelike Ricci curvature from order-theoretic data.
Research shows no dual solutions for Lorentzian cost functions in general, but proves their existence under certain conditions.
problem Existence of dual solutions for Lorentzian cost functions in optimal transportation problems.
method Analyzes dual problem in Lorentz-Finsler geometry, proves existence under natural assumptions, and shows implications for optimal transport.
result Existence of dual solutions under specific conditions, implying timelike optimal transport on a set of full measure.
Sharp inequality for Lorentzian spaces with timelike Ricci bounds.
problem Establishing bounds on achronal hypersurfaces in Lorentzian spaces.
method Optimal transport and synthetic TCDpe(K,N) spaces. result Sharp isoperimetric-type inequality for Lorentzian spaces.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
Survey on a nonlinear d'Alembertian from general relativity.
problem Understanding a new nonlinear operator from relativity.
method Review of a new distributional d'Alembertian, comparison estimates, and exact representation formulas.
result Control of the timelike cut locus through optimal transport.
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with C1,1 regularizing pairs method Local semiconcavity and future-directed timelike superdifferentials
result Derives C1,1 regularizing pairs for optimal transport under general assumptions Entropy convexity characterizes strong energy condition in spacetimes.
problem Characterizing strong energy condition in nonsmooth spacetimes.
method Lifting fractional powers of Lorentz distance to probability measures and showing geodesic convexity of Boltzmann-Shannon entropy.
result Strong energy condition is equivalent to geodesic convexity of Boltzmann-Shannon entropy.
Study introduces new curvature conditions for Lorentzian spaces using Rényi entropy.
problem Developing synthetic curvature conditions for Lorentzian spaces.
method Introducing timelike curvature-dimension conditions and measure-contraction properties using Rényi entropy.
result Equivalence of new curvature conditions to entropic counterparts.
We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…
A function that optimally aligns a timelike vector field with its gradients
problem Finding a time function that aligns a timelike vector field with its gradients
method Introducing a functional that penalizes null gradients and minimizes misalignment
result Proving the existence of a unique alignment time function under suitable conditions
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally maximizing. Second, we obtain a parametrization of timelike, spacelike, lightlike norma…
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the M…
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
Paper classifies timelike Thomsen surfaces and reveals their geometric properties.
problem Classifying timelike Thomsen surfaces and understanding their geometric properties.
method Use of Lorentz conformal and null coordinates, representation theorems, and geometric invariants.
result Revealed the relationship between timelike Thomsen surfaces and timelike minimal surfaces with planar curvature lines.
Paper classifies timelike Bonnet surfaces in Lorentzian 3-manifolds.
problem Classifying timelike Bonnet surfaces in Lorentzian 3-manifolds.
method Provided a necessary and sufficient condition for timelike Bonnet surfaces in Lorentzian 3-spaces.
result Theorem classifying timelike Bonnet surfaces in a Lorentzian 3-space.
In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the r…
Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong q-timelike Brunn-Minkowski condition and proving equivalence to curvature conditions. result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
This study examines geometric properties and offsets of slant timelike-ruled surfaces.
problem Geometric properties and offsets of slant timelike-ruled surfaces in Minkowski 3-space.
method Derivation of parametric formulation, conditions for coaxial alignment, examination through Blaschke and Darboux frames.
result Conditions ensuring the coaxial alignment of the central normal with the ruling direction of the offset surface.
The study classifies timelike meridian surfaces in Minkowski 4-space.
problem Characterizing timelike meridian surfaces in Minkowski space.
method Analyzing surfaces as one-parameter systems of meridians of rotational hypersurfaces.
result Classification of timelike meridian surfaces with various curvature properties.
The paper studies timelike loxodromes on specific Lorentzian helicoidal surfaces.
problem Analyzing timelike loxodromes on Lorentzian helicoidal surfaces.
method First-order differential equations, general solutions, explicit parametrizations.
result Explicit parametrizations of timelike loxodromes on Lorentzian helicoidal surfaces.
Paper proves timelike minimal surfaces with any number of ends exist.
problem Existence of timelike minimal surfaces with arbitrary ends.
method Analyzes asymptotic behavior and topology of constructed surfaces.
result Timelike minimal surfaces with arbitrary ends exist.
Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
problem Characterizing the geometry of spherical simplices.
method Proved isometry to vector spaces with a timelike norm.
result Timelike spherical Hilbert geometry of simplices is isometric to a union of six copies of vector spaces.
New methods estimate transport-growth pairs in unbalanced optimal transport.
problem Statistical guarantees for Monge-type estimation in unbalanced optimal transport remain limited.
method Developed two estimators for transport-growth pairs under different setups.
result Achieved minimax optimal rate for estimation of transport-growth pairs.
Novel approach learns optimal transport using convex neural networks.
problem Learning optimal transport between distributions from samples.
method Solving a minimax optimization to learn two convex functions, representing the optimal transport map.
result The approach finds optimal transport mappings that are independent of initialization and can handle discontinuous distributions.
The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
New method finds closed timelike geodesics on Lorentzian manifolds.
problem Existence of closed timelike geodesics in Lorentzian geometry.
method Introducing timelike geodesic homotopy and combining with a local length argument.
result Provides new results on the existence of closed timelike geodesics.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
We introduce a new approach to the study of timelike minimal surfaces in the Lorentz-Minkowski space through a split-complex representation formula for this kind of surface. As applications, we solve the Björling problem for timelike surfaces and obtain interesting examples and related results. Using the Björling repre…
Introduces statistical optimal transport for probabilistic lectures.
problem No specific problem stated; focuses on introduction.
method Lecture-based introduction to statistical optimal transport.
result Provides an introduction to statistical optimal transport.
Study timelike meridian surfaces in Minkowski 4-space with specific properties.
problem Characterize timelike meridian surfaces with special properties in Minkowski 4-space.
method Analyze different classes of timelike meridian surfaces with constant Gauss curvature, mean curvature, and parallel normalized mean curvature vector field.
result Explicit solutions to PDEs describing timelike surfaces with parallel normalized mean curvature vector field.
Maximal surfaces in L3 correspond to timelike minimal surfaces.
problem Establishing a correspondence between maximal and timelike minimal surfaces in L3. method Linear transformation between maximal surfaces and timelike minimal surfaces, preserving singularities and Gauss map.
result One-one correspondence and preservation of properties between maximal and timelike minimal surfaces.
Study timelike minimal surfaces in Heisenberg group using harmonic maps.
problem Characterize timelike minimal surfaces in the Heisenberg group.
method Characterization through non-conformal Lorentz harmonic maps and generalized Weierstrass type representation.
result Established generalized Weierstrass type representation for timelike minimal surfaces.
Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.
problem Characterize minimal timelike surfaces in R13. method Use a Weierstrass-type formula with holomorphic functions in split-complex numbers to find canonical parameters and corresponding holomorphic functions.
result Enneper surfaces are the only minimal timelike surfaces with polynomial parametrization of degree 3 in isothermal parameters.
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
Constructs Lorentzian harmonic maps and associated timelike surfaces.
problem Lorentzian harmonic maps and timelike surfaces properties.
method Constructs framed null curves and solves eigenvalue equation.
result Characterizes singularities on timelike minimal surfaces.
A new spinor approach proves properties of timelike surfaces in Minkowski space.
problem Characterizing timelike surfaces in 4D Minkowski space.
method Spinor field and Dirac equation on timelike surfaces.
result Spinor representation formula connects surface properties to its immersion.
Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.
problem Exploring singularities of timelike minimal surfaces in Minkowski 3-space.
method Existence and non-existence theorems, criteria for specific singularities.
result Various singularities unique to timelike minimal surfaces, including cuspidal butterfly and (2,5)-cuspidal edge. NOT learns optimal transport plans, kernel costs improve performance.
problem NOT algorithm learns non-optimal plans with weak quadratic costs.
method Introduced kernel weak quadratic costs to improve NOT's performance.
result Kernel costs provide improved theoretical and practical guarantees.
New algorithm solves unbalanced optimal transport on trees in quasi-linear time.
problem Efficiently solving unbalanced optimal transport problems on trees.
method Proposed an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric.
result Solves unbalanced optimal transport on trees in quasi-linear time (less than one second for a tree with one million nodes).
Deep learning models optimize urban transportation scheduling.
problem Optimizing transportation systems with complex dynamics and large data sets.
method Developed deep learning metamodels for simulators and reinforcement learning algorithms.
result Improved optimal scheduling of travelers on transportation networks.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.