We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.
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The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.
problem Volume comparison in Lorentzian pre-length spaces.
method Introducing modified timelike Hausdorff measures and establishing volume comparison inequalities using timelike Lipschitz maps.
result Coincidence of modified and original volume measures on smooth spacetimes and some pre-length spaces.
We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.
The paper studies marginally trapped submanifolds in Lorentzian manifolds under null energy condition.
problem Understanding marginally trapped submanifolds in Lorentzian manifolds.
method Analyzes properties of marginally trapped submanifolds in a Lorentzian manifold satisfying the null energy condition.
result Marginally trapped submanifolds have locally volume-maximizing properties in certain null hypersurfaces.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to -dimensional Lorentz--Finsler manifolds.
result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial or…
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.
Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…
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.
In the lorentzian product we give a comparison between the -volume of an entire -maximal graph and the -volume of the hyperbolic under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theor…
The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact,…
Study establishes time functions in Lorentzian spaces without requiring manifold structure.
problem Existence and properties of time functions in Lorentzian spaces.
method Characterization of time functions by K-causality, modified volume functions, and global hyperbolicity.
result No manifold structure is needed for suitable time functions in Lorentzian spaces.
Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable a…
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applicat…
This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the exi…
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local v…
Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebr…
The paper estimates area and volume for spacetimes with integral mean curvature bounds.
problem Estimating area and volume for spacetimes with specific curvature conditions.
method Using strong energy condition and norms of second fundamental form/mean curvature.
result Established area and volume estimates for spacetimes.
Study sequences of static spacetimes using null distance convergence.
problem How to define convergence for sequences of spacetimes.
method Define null distance metric space structure compatible with Lorentzian structure.
result Prove VADB theorem for sequences of static spacetimes with null distance.
In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. T…
Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric,…
A singularity theorem based on asymptotic volume growth
problem Proving singularity theorems
method Introducing asymptotic volume-expansion invariants
result Proving an explicit upper bound on the time-separation from a hypersurface to its chronological past
We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2-torus, we prove that a metric with constant cu…
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow…
Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.
problem Geometric and topological constraints on codimension-two spacelike submanifolds.
method Analysis of submanifolds with umbilical lightlike normal directions, using geometric and topological constraints.
result Any such submanifold is contained in a lightlike hypersurface, which is totally umbilical if the lightlike normal direction is umbilical.
We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter be…
Proves Hawking's theorem for less smooth spacetime metrics.
problem Proving Hawking's singularity theorem for less smooth spacetime metrics.
method New estimates for Ricci curvature and a segment-type inequality for volume control.
result Proves Hawking's singularity theorem for Lipschitz metrics.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
Defines metrics for Lorentzian spaces and explores maximal developments.
problem Defining metrics and maximal developments in Lorentzian spaces.
method Defining Lorentzian spaces and using functorial properties.
result Explicit non-spacetime example of maximal globally hyperbolic Lorentzian space.
Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
problem Global optimality of extremal trajectories in a series of Lorentzian structures.
method Analysis of a one-parametric series of left-invariant Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
result Properties of the Lorentzian structures deform to those of the sub-Lorentzian structure in a limit case.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …
Study gluing of Lorentzian length spaces and their causal ladder properties.
problem Compatibility of Lorentzian amalgamation with length space properties.
method Conditions for gluing Lorentzian length spaces and criteria for causal ladder preservation.
result Gluing of Lorentzian length spaces yields again a Lorentzian length space under certain conditions.
Defines doubling conditions for Lorentzian spaces linked to curvature bounds.
problem Relating doubling conditions to curvature bounds in Lorentzian geometry.
method Defines doubling conditions using chronological diamonds and proves implications with timelike curvature bounds.
result Relates doubling to curvature bounds in Lorentzian geometry.
Research explores Lorentzian distances on a specific geometric plane.
problem Investigating Lorentzian structures on a 2D geometric plane.
method Analyzes sectional curvature, attainable sets, and Lorentzian length maximizers.
result Describes distance properties and spheres in the context of Lorentzian geometry.
Finds a limit on Lorentzian manifolds.
problem Limiting the number of Lorentzian manifolds.
method Functor from Lorentzian to Riemannian manifolds.
result Finiteness result on Lorentzian manifolds.
The study proves a transverse diameter theorem for Lorentzian foliations.
problem Understanding the geometry of foliations in Lorentzian spacetimes.
method Developed a novel causality structure on leaf spaces via transverse Lorentzian geometry.
result Derived a transverse diameter theorem for Lorentzian foliations and orbifolds.
The paper classifies 3D Lorentzian Lie groups.
problem Classifying 3D Lorentzian Lie groups.
method Complete classification through mathematical analysis.
result Three-dimensional Lorentzian Lie groups have been completely classified.
Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.
problem Characterizing singularities on timelike minimal surfaces.
method Constructing timelike minimal surfaces as Lorentzian harmonic maps and analyzing their singularities.
result Criteria for cuspidal edges, swallowtails, and cuspidal cross caps are provided.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.
Compact group found for special Lorentzian manifolds.
problem Understanding the conformal group of specific Lorentzian manifolds.
method Analyzing real analytic Lorentzian manifolds with finite fundamental group.
result The conformal group is compact for these manifolds.
Study on Lorentzian spaces with curvature bounds, proving comparison theorems.
problem Understanding curvature bounds in Lorentzian spaces.
method Introduced normalized angle for Lorentzian pre-length spaces, proving comparison theorems.
result Established local Lorentzian Toponogov theorem and Alexandrov convexity property.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
Topology change and selection rules for high-dimensional $\Spin(1, n)_0$-Lorentzian cobordismsmath.GT
We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…