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.
Proves density and mass theorems for specific initial data sets.
problem Initial data sets with boundary in spacetime.
method Harmonic asymptotics and dominant energy condition.
result Spacetime positive mass theorem for initial data sets with apparent horizon boundary.
When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …
Constructs constant spacetime mean curvature surfaces for hyperboloidal initial data sets.
problem Creating a foliation of constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets.
method Long time limit of volume preserving spacetime mean curvature flow starting from a constant mean curvature foliation.
result Obtains a foliation of constant spacetime mean curvature surfaces as the long time limit.
Estimates bandwidth for CMC initial data sets.
problem Estimating bandwidth for constant mean curvature (CMC) initial data sets.
method Three independent proofs: stability of null expansion, spacetime harmonic function perturbation, Dirac operator.
result Generalized Gromov's band width estimate to CMC initial data sets.
Smooth dec initial data sets may not extend to smooth spacetimes.
problem Whether every dec initial data set can be extended to a smooth spacetime.
method Examined the converse of the dominant energy condition for initial data sets and spacelike hypersurfaces.
result Not all dec initial data sets can be extended to smooth spacetimes.
Proves critical points of ADM mass correspond to specific initial data sets.
problem Finding initial data sets with fixed Bartnik boundary data.
method Proves existence of critical points on a Banach manifold.
result Critical points of ADM mass correspond to initial data sets with generalized Killing vector fields.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets
problem Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets
method Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets
result Prove area-charge inequalities for free boundary MOTS in initial data sets for the Einstein-Maxwell equations with vanishing magnetic fields
Initial data with zero mass must be in pp-wave spacetimes.
problem Proving initial data with zero mass must be in pp-wave spacetimes.
method Spinorial methods combined with spacetime harmonic functions.
result Initial data with zero mass must be contained in pp-wave spacetimes.
Proves principles and estimates for initial data sets in Einstein equations.
problem Understanding initial data sets in Einstein equations.
method Spinorial Callias operator approach.
result Proves long neck principle and width estimates.
Study proposes initial data sets for solving gravitational equations, proving energy estimates.
problem Solving the constraint equations in the evolutionary form.
method Proposes a family of initial data sets, proving Penrose-like energy estimates.
result Established existence of solutions for specific cases.
Proves rigidity for specific initial data sets under the dominant energy condition.
problem Rigidity of initial data sets with boundary and convex polytopes.
method Solution of boundary value problems for Dirac operators and approximations by manifolds with smooth boundary.
result Proves rigidity for compact smooth spin manifolds and convex polytopes under the dominant energy condition.
The paper proves rigidity results for compact initial data sets.
problem Understanding the structure of compact initial data sets.
method Proving rigidity results under specific conditions.
result Global version of the main result in [15] is obtained.
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.
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…
Constructs initial data for Einstein vacuum equations involving multiple localized gravitational sources.
problem Modeling the interaction of distant gravitational systems in general relativity.
method Time-symmetric initial data construction using gluing schemes and localized sources.
result Produces initial data sets with finite ADM mass and multiple Einstein-Rosen bridges.
Initial data for pp-wave spacetimes constructed in 4D.
problem Characterizing initial data for pp-wave spacetimes. method Constructs a vacuum initial data set with extra conditions related to CKID.
result Data development is a subset of a vacuum pp-wave. Study finds rigid properties of boundary-free hypersurfaces in specific data sets.
problem Rigidity of free boundary hypersurfaces in initial data sets with boundary.
method Extending local splitting theorems and applying results on free boundary MOTS.
result Rigidity results for compact free boundary hypersurfaces in initial data sets with boundary.
We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely …
Proves positive mass theorem for spin initial data sets with arbitrary ends and dominant energy shields.
problem Proving the positive mass theorem for spin initial data sets with various ends and energy shields.
method Modification of Witten's approach involving an additional independent timelike direction in the spinor bundle.
result Positive mass theorem for spin initial data sets with arbitrary ends and dominant energy shields.
Article strengthens initial data rigidity theorem to show unique spacetime extension.
problem Initial data rigidity in spacetime geometry.
method Showed initial data sets carry a lightlike parallel vector field, leading to unique spacetime extension.
result Local uniqueness of spacetimes extending initial data sets under dominant energy condition.
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.
We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No…
Paper proves new inequalities for Einstein-Maxwell data sets.
problem Establishing area-charge inequalities for Einstein-Maxwell initial data sets.
method Applying Gromov's μ-bubble technique in a new geometric context.
result Novel rigidity theorems for noncompact Einstein-Maxwell data sets.
In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is ra…
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…
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Rie…
The paper proves positive energy-momentum theorems for charged AdS initial data sets.
problem Proving positive energy-momentum theorems for charged asymptotically AdS initial data sets.
method Introducing a charged energy-momentum functional and establishing positive theorems under a dominant energy condition.
result The charged energy-momentum functional is non-negative on a natural real cone.
We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conf…
We carry out "exotic gluings" a la Carlotto-Schoen for asymptotically hyperbolic general relativistic initial data sets. In particular we obtain a direct construction of non-trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity.
The study extends conserved quantities theory to non-compact boundary initial data sets.
problem Extending conserved quantities theory to initial data sets with non-compact boundaries.
method Analysis of scalar curvature and mean curvature in the interior and boundary.
result Rigidity/flexibility phenomena in positive mass theorems and Penrose inequalities.
Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.
problem Positivity and rigidity of Hawking quasi-local energy in asymptotically flat spacetimes.
method Lyapunov-Schmidt reduction within a Willmore-foliation framework.
result Existence and uniqueness of foliations by Hawking surfaces, positivity and large-sphere limit of Hawking energy.
Paper extends positive energy theorem to anti-de Sitter spacetimes.
problem Proving positive energy theorem for weighted anti-de Sitter spacetimes.
method Generalized positive energy theorem for 3D anti-de Sitter initial data sets.
result Positive energy theorem proved for weighted anti-de Sitter spacetimes.
Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.
problem Proving the spacetime positive mass theorem for specific spacetime configurations.
method Solving a mixed boundary value problem for the Dirac-Witten operator with a Callias potential.
result Established spacetime positive mass theorem for asymptotically flat spin initial data sets with arbitrary ends.
The paper proves energy theorems for specific initial data sets in 3D spacetime.
problem Establishing energy theorems for specific initial data sets in 3D spacetime.
method Analysis of level sets of spacetime harmonic functions.
result Rigidity results showing vanishing total energy imply isometric manifolds.
Study examines stability of MOTS in symmetric initial data sets.
problem Stability of marginally outer trapped surfaces in symmetric initial data sets.
method Characterizes instability through vector decomposition and properties of normal and tangent components.
result Characterizes instability of MOTS by the nature of zero sets and divergences.
New method constructs flat initial data for Einstein's equations.
problem Constructing asymptotically flat initial data for Einstein's equations.
method Explicit solution operators with localization properties.
result Improved decay rate and nontrivial initial data construction.
Proves Penrose inequality for cohomogeneity one initial data sets.
problem Proving Penrose inequality for specific initial data sets.
method Analyzing asymptotically flat and hyperbolic initial data sets under cohomogeneity one actions.
result Total mass is bounded below by a function of outermost apparent horizon area, with equality for Schwarzschild(-AdS) embeddings.
The paper proves a spacetime positive mass theorem for singular initial data sets.
problem Proving the positive mass theorem for initial data sets with corners.
method Extending Hirsch-Kazaras-Khuri's method to singular cases using Hirsch-Miao-Tsang ideas.
result Integral lower bound on spacetime mass and characterisation of zero mass.
This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students who have taken a first course in general relativity, the article first discusses …
Constructs initial data for Einstein equations and estimates Bartnik mass outside time-symmetry.
problem Estimating Bartnik mass outside time-symmetry.
method Constructs initial data for Einstein equations and connects Bartnik data to time-symmetric data.
result Obtains estimates for the Bartnik mass outside of time-symmetry.
In this note we study the topology of 3-dimensional initial data sets with horizons of a sort associated with asymptotically locally anti-de Sitter spacetimes. We show that, within this class, those initial data sets which contain no (immersed) marginally outer trapped surfaces in their interior must have simple topolo…
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.
Positive energy theorems for spin initial data with charge in higher dimensions.
problem Establishing positive energy theorems for spin initial data with charge in dimensions n≥4. method Using a dominant energy condition and asymptotically flat ends, extending classical theorems.
result Extending classical positive energy theorems to spin initial data with charge.
This work removes logarithmic singularities from hyperboloidal initial data without creating new ones.
problem Logarithmic singularities in hyperboloidal initial data sets.
method Evolutionary framework of the constraint equations and generalization of Beyer and Ritchie's result.
result Generic solutions of the constraint equations are free of logarithmic singularities.
We find necessary and sufficient conditions ensuring that the vacuum development of an initial data set of the Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID) and they extend the well-known Killing initial data (KID) that have been know…