We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data c…
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48 results for “trapping sets”
The study generalizes Santalo's formula and shows stability of trapping sets in Riemannian manifolds.
problem Stability of trapping sets in Riemannian manifolds under smooth perturbations of obstacles.
method Generalization of Santalo's formula applied to billiard trajectories in the exterior of obstacles.
result The measure of the set of trapped points depends continuously on perturbations of the obstacle.
Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.
problem Understanding marginally trapped surfaces in perturbed Schwarzschild spacetime.
method Developed a method to study spacelike surfaces in a double null coordinate system.
result For every incoming null hypersurface nearly spherically symmetric, there exists a unique embedded marginally trapped surface.
Study shows certain manifolds uniquely determined by X-ray data.
problem Determining manifolds from X-ray data.
method Injectivity of X-ray transform over symmetric solenoidal 2-tensors.
result Smooth compact manifolds with strictly convex boundary, no conjugate points, and hyperbolic trapped set are locally marked boundary rigid.
Proves stable smooth minimal hypersurface for trapped region boundary in large dimensional manifolds.
problem Boundary stability of trapped region in large dimensional manifolds.
method Analyzes asymptotically Euclidean Riemannian manifolds of dimension at least 3.
result Boundary is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.
This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…
Global and uniqueness properties of stationary and static spacetimes with outer trapped surfacesgr-qc
Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null energy condition, the future Cauchy development of the data is a black hole spa…
In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable margi…
Study of trapped photons in Kerr spacetime's phase space.
problem Characterize trapped photons in Kerr spacetime.
method Explicit proof and new proof of trapped photons' set as a smooth 5D submanifold.
result Set of trapped photons forms a smooth 5D submanifold with topology .
Study stability and rigidity of axisymmetric marginally outer trapped surfaces.
problem Stability and rigidity of axisymmetric marginally outer trapped surfaces.
method Refined results from initial data sets with Killing vector fields, using new foliation lemma.
result Conditions for the stability of axisymmetric MOTS and new foliation lemma.
Injectivity of X-ray transform on solenoidal tensors proven for certain manifolds.
problem Injectivity of X-ray transform on solenoidal tensors on manifolds with hyperbolic trapped set.
method Proof of equivalence principle concerning injectivity and surjectivity of operators on symmetric solenoidal tensors.
result Injectivity of X-ray transform on solenoidal tensors of any order for surfaces with hyperbolic trapped set.
Study shows genericity of singularities in spacetimes with weakly trapped submanifolds.
problem Ensuring nonspacelike geodesic incompleteness in spacetimes with weakly trapped submanifolds.
method Use strong Whitney topologies on spaces of Lorentzian metrics and Hilbert manifold structures on initial data sets to prove genericity of null geodesic incompleteness.
result The phenomenon of nonspacelike geodesic incompleteness is generic in a precise technical sense.
Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static situations. In this paper we study an important aspect of this equivalence, namely wheth…
We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfa…
Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem…
Deep learning system speeds up wildlife species identification from camera trap images.
problem Manual review of camera trap images is slow and resource-intensive.
method Combines machine and human intelligence for active learning.
result Matches state-of-the-art accuracy with minimal manual labels.
Study shows no closed trapped submanifolds can be tangent to certain spacelike hypersurfaces.
problem Existence of closed trapped submanifolds in spacetime regions foliated by specific hypersurfaces.
method Introduced future convex spacelike/null hypersurfaces and proved no dimensional closed trapped submanifolds can be tangent to these hypersurfaces from their future side.
result Closed trapped submanifolds cannot be found in open spacetime regions foliated by future convex hypersurfaces.
Study analyzes household capital risk and poverty trapping, deriving a new function for capital deficit distribution.
problem Analyzing the risk of household capital falling into poverty.
method Introduced a new Gerber-Shiu function to model trapping time and capital deficit distribution.
result Derived a model for capital deficit distribution at trapping using GB distributions.
Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds i…
New method constructs spacelike data leading to trapped surfaces.
problem Formation of trapped surfaces from spacelike initial data.
method Free data formalism and local existence result.
result Data can be extended to asymptotically flat Cauchy data.
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increas…
Rigidity results for initial data sets related to the positive mass theorem.
problem Rigidity of initial data sets in general relativity.
method Establishing conditions for weak outermost marginally outer trapped surfaces and rigidity results for Riemannian manifolds.
result Marginally outer trapped surfaces are weakly outermost under certain conditions.
Paper studies stochastic optimization methods with momentum, proving convergence and avoiding traps.
problem Optimizing non-convex functions with momentum.
method Unified analysis of stochastic gradient descent variants, including S-NAG and Adam.
result Convergence to critical points and avoidance of undesired critical points like local maxima or saddle points.
New algorithm helps avoid traps in reinforcement learning.
problem Existing reinforcement learning bounds assume no traps, limiting applicability.
method Introduces DRL (delegative reinforcement learning) allowing occasional advisor input.
result Derives a new regret bound without assuming episodic or trap-free environments.
Analytic metrics are uniquely determined by their scattering map.
problem Determining Riemannian manifolds from scattering data.
method Analytic negatively curved Riemannian manifolds with strictly convex boundary.
result The scattering map determines the manifold up to isometry.
Paper proves rigidity of initial data sets with boundary and capillary MOTS.
problem Rigidity of initial data sets with boundary and capillary MOTS.
method Estimates area of MOTS, proves rigidity for 3D, extends to high dimensions using Yamabe constant.
result Rigidity results for initial data sets with boundary and capillary MOTS.
The paper introduces canonical parameters for marginally trapped surfaces in Minkowski space.
problem Determining marginally trapped surfaces in Minkowski space.
method Introducing canonical parameters and proving existence and uniqueness theorems.
result Every marginally trapped surface is determined by three smooth functions.
Generic singularities found in spacetimes with weakly trapped submanifolds.
problem Existence of singularities in spacetimes with weakly trapped submanifolds.
method Using Whitney topologies and singularity theorems, the study shows genericity of singularities in spacetimes with weakly trapped submanifolds.
result Generic singularities are found in spacetimes with weakly trapped submanifolds.
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by , where is a lower bound on a natural energy-momentum term. We then consider th…
Study shows how charged MOTS restrict spacetime configurations.
problem Understanding charged initial data sets and their rigidity.
method Analyzes marginally outer trapped surfaces with charge considerations.
result New rigidity results for charged initial data sets.
Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
Wariness and Poverty Trapsq-fin.CP
Wariness affects poverty traps and equilibrium diversity in economic models.
problem The impact of wariness on poverty traps and equilibrium diversity in economic models.
method An overlapping generations model to explore the effects of wariness on poverty traps and equilibrium diversity.
result Wariness can amplify or mitigate the likelihood of poverty traps and can lead to multiple intertemporal equilibria.
Researchers extend Chen, Erchenko, and Gogolev's result to more cases.
problem Embedding manifolds with hyperbolic geodesic trapped sets into compact manifolds with Anosov geodesic flows.
method Explains how assumptions can be removed to apply the result to all reasonable 3D examples.
result A broader applicability of the original result to all reasonable 3D examples.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…
Proves two non-trapping obstacles coincide if scattering rays have similar travelling times or scattering length spectra.
problem Identifying non-trapping obstacles based on scattering properties.
method Proves two obstacles coincide if their scattering rays have similar travelling times or scattering length spectra under weak non-degeneracy conditions.
result Two non-trapping obstacles coincide if their scattering rays have similar travelling times or scattering length spectra.
Study of trapped submanifolds in null hypersurfaces with curvature constraints.
problem Understanding trapped submanifolds in null hypersurfaces with curvature constraints.
method Analyzing null hypersurfaces with constant sectional curvature and proving properties of trapped submanifolds.
result Null trapping horizons cannot exist in null hypersurfaces with non-positive curvature.
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 …
Paper proves trapped surface formation for EMCSF system without symmetry assumptions.
problem Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system.
method Scale-critical trapped surface formation result established from past null infinity.
result Focusing of gravitational waves, concentration of electromagnetic fields, or condensation of scalar fields can lead to trapped surface formation.
Overview of marginally trapped surfaces in various spacetimes.
problem Understanding marginally trapped surfaces in different spacetimes.
method Differential geometric study of marginally trapped surfaces in Minkowski, de Sitter, anti-de Sitter, and Robertson-Walker spacetimes.
result General local descriptions and classifications of these surfaces.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov norm and extending to smallness.
result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
Study shows IMP gluing spacetimes are incomplete.
problem Local geometry and completeness of IMP gluing spacetimes.
method Investigation of IMP gluing initial data sets, existence of outer trapped surfaces, and application of Penrose's incompleteness theorem.
result Implication of null incompleteness for IMP gluing spacetimes.
Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an ingredient in the pro…
The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using t…
We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay p…
The paper proves a Penrose inequality involving quasi-local mass and outer trapped surfaces.
problem Detecting and quantifying the presence of black holes in spacetime.
method Comparison theorem and quasi-local mass analysis.
result Established a Penrose inequality linking quasi-local mass and outer trapped surfaces.
Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.