We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of -regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for -metrics, an…
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48 results for “Hawking-Penrose singularity”
We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology.
We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature in timelike directions. We prove that they still hold when is finite, and when is infinite, they hold under the additional assumptio…
Researchers prove a 30-year-old cosmological conjecture about spacetime.
problem The rigidity of the cosmological Hawking--Penrose singularity theorem.
method Combining global viscosity solutions and elliptic approaches.
result A timelike geodesically complete spacetime splits isometrically as a Lorentzian product.
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.
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 analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions , including all negative synthetic dimensions. The rigidity of the timelike spli…
Proves a conjecture about complete convex surfaces containing an umbilic point.
problem Proving a conjecture about complete convex surfaces.
method Indirect proof using Riemann-Hilbert boundary value problems and existence results for holomorphic discs.
result Proves the Toponogov conjecture on complete convex planes.
The Bakry-Emery generalized Ricci tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame. Recent results from the mathematics literature show that standard singularity and splitting theorems that hold when an energy condition is applied in general relativity also hold when …
In the early 80's S.-T. Yau posed the problem of establishing the rigidity of the Hawking-Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets - the horospheres. The regularity theory in the Lorentzian …
In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still …
Unified approach to various energy conditions in spacetime geometry.
problem Synthetic quantification of energy conditions in spacetime.
method Introducing entropic timelike curvature dimension condition with variable Ricci curvature bounds.
result Unified approach to various energy conditions including strong, weak, and null energy conditions.
New metrics derived from geodesics simplify semi-Riemannian geometry.
problem Simplifying complex semi-Riemannian metrics.
method Constructing plane wave limits along geodesics.
result Generalizes Penrose's limit and encodes tensorial geometry.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under -limits.
method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
New singularity concept in GR: volume singularities.
problem Understanding spacetime singularities in General Relativity.
method Introducing a new type of singularity: volume singularities.
result Volume singularities are hidden by event horizons.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
Distance squared functions on singular surfaces parameterized by smooth maps -equivalent to , , and math.DG
Study describes singularities of distance squared functions on singular surfaces.
problem Characterizing singularities of distance squared functions on singular surfaces.
method Using smooth map-germs , , , and singularities, the study describes singularities via blowing-ups.
result Characterization of singularities of wave-fronts and caustics of singular surfaces.
In this paper, we define the set of singular grid diagrams which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on which induce the bijection onto e…
Positive mass theorem for asymptotically flat spin manifolds with isolated conical singularitiesmath.DG
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a -cuspidal edge is considered.
Rectifies flat singular points of area-minimizing currents with singularity degree > 1.
problem Rectifying flat singular points of area-minimizing currents with singularity degree > 1.
method Subdividing singular points based on singularity degree and proving rectifiability of points with singularity degree > 1.
result The set of points with singularity degree > 1 is (m-2)-rectifiable.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper studies singularities of pedal curves of hyperbolic frontals.
problem Investigating singularities of pedal curves of spacelike frontals in hyperbolic 2-space.
method Analyzing singularities of pedal curves based on dual curve germs and pedal point locations.
result The singularities of pedal curves depend on the singularities of the first hyperbolic Legendrian curvature germ and the pedal point for non-singular dual curve germs. For singular dual curve germs, additional dependence on both Legendrian curvature germs is observed.
The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
Maxfaces can have cuspidal edges near certain singularities.
problem Characterizing singularities on maxfaces.
method Analyzing singular Björling data and proving geometric properties.
result Near a maxface with a specific type of singularity, there exists another maxface with a cuspidal edge.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.
problem Understanding geodesic behavior near singularities in Riemannian manifolds.
method Analytical study of geodesics on Riemannian manifolds near singularities.
result The winding number of geodesics around a singularity depends on the singularity type and approaches infinity as the singularity becomes cuspidal.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical singularities of planar…
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and -tangent bundles approach to resolve singularities.
result Established a Poincaré-Hopf theorem for -tangent bundles.
Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
Curvature criteria for A-simple singularities and their parallel curves identified.
problem Determining singularity types of A-simple singularities and their parallel curves.
method Defined curvature parameters and criteria for A-simple singularities.
result Criteria to determine singularity types of A-simple singularities and their parallel curves.
We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler's program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse-Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its sy…
Algorithm classifies saddle-focus singularities in Hamiltonian systems.
problem Classifying nondegenerate saddle-focus singularities in integrable Hamiltonian systems.
method Developed an algorithm based on semi-local equivalence to represent singularities as almost direct products.
result Obtained complete lists of saddle-focus singularities of complexities 1, 2, and 3.
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Study on oriented disingquandles for distinguishing singular links.
problem Distinguishing singular links of different colors.
method Introduced oriented dichromatic singular links and oriented disingquandles to define invariants.
result Found invariants that can distinguish some singular links.
Study singularities of CMC 1 surfaces in de Sitter space.
problem Characterize singularities of spacelike CMC 1 surfaces in de Sitter space.
method Proved duality between singular points and introduced invariants.
result Classified non-degenerate singular points on spacelike CMC 1 surfaces.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
Small time heat expansion of the laplacian on an analytic hypersurface with an isolated singularitymath.AP
Proves heat expansion for Laplacian on a singularity.
problem Analytic hypersurface with isolated singularity and Laplacian heat expansion.
method Local parametrization, Newton scheme, quasihomogeneous tangent cone, local models with irregular singularities.
result Existence of small time heat expansion for Laplace operator.
New Alexander polynomial for singular knots improves upon existing methods.
problem Defining a polynomial invariant for singular knots.
method Introducing a perturbed Alexander polynomial.
result The new polynomial agrees with previous definitions for long knots.