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.
Transforms hyperbolic to flat data, deriving geometric inequalities.
problem Deriving geometric inequalities in asymptotically AdS hyperbolic spacetimes.
method Constructs transformations preserving physical quantities to relate hyperbolic to flat spacetimes.
result Derives geometric inequalities from flat counterparts.
We study Ricci flows on Rn, n≥3, that evolve from asymptotically flat initial data. Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is constant in time along the flow. We then specialize to the case of rotationally symmetr…
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.
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.
The paper proves positive mass theorems for initial data sets with noncompact boundaries.
problem Proving positive mass theorems for initial data sets with noncompact boundaries.
method Defining an energy-momentum vector at spatial infinity and proving positive mass inequalities under DECs.
result Proves positive mass inequalities for initial data sets with noncompact boundaries under suitable DECs.
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.
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.
Proves positive mass theorem for specific initial data sets with corners.
problem Initial data sets with corners and non-smooth boundaries.
method Axially symmetric, maximal, complete initial data sets with two ends, proving for axially symmetric, simply connected, maximal, complete initial data sets with two ends.
result Proves positive mass theorem for specific initial data sets with angular momentum and charges.
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 in all dimensions.
problem Proving the spacetime positive mass theorem in arbitrary dimensions.
method Using Brendle--Wang's Riemannian positive mass theorem approach.
result Proves the spacetime positive mass theorem for all dimensions.
Developed tools to compute charged Bartnik mass for Einstein-Maxwell equations.
problem Computing quasi-local mass for charged initial data sets.
method Created extensions and gluing techniques for time-symmetric initial data sets of Einstein-Maxwell equations.
result Computed ad-hoc charged Bartnik mass for suitable charged minimal Bartnik data.
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…
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
Proves positive mass theorem for hyperbolic manifolds with ends.
problem Establishing positive mass theorem for complex initial data sets.
method Used spectral PSC, Jang equation, and quantitative shielding theorem.
result Proved positive mass theorem for asymptotically hyperbolic manifolds.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
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.
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
Affirms rigidity conjecture for spacetime positive mass theorem in dimensions less than eight.
problem Proving the rigidity conjecture for the spacetime positive mass theorem in dimensions less than eight.
method Analyzing asymptotically flat initial data sets with dominant energy condition and E=∣P∣. Removing dimensional restriction with positive mass inequality assumption. result Affirmation of the rigidity conjecture for spacetime positive mass theorem in dimensions less than eight.
We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initi…
Proves path connectedness of asymptotically flat metrics with boundary.
problem Proving path connectedness of asymptotically flat metrics with boundary.
method Generalization of Marques' result to compact manifolds with boundary, differential topology, and a new proof.
result Space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on R^3\B^3 is path connected.
New approach removes obstructions in gluing spacelike and null hypersurfaces in Einstein equations.
problem Gluing two solutions of the Einstein equations along a hypersurface.
method Active utilization of nonlinearity, low-frequency linear analysis, high-frequency nonlinear control.
result Removes 10-dimensional obstructions in null and spacelike gluing problems.
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.
Paper proves a generalized Penrose conjecture for flat initial data.
problem Proving the generalized Penrose conjecture for flat initial data.
method Developed a new geometric evolution called the σ-inverse mean curvature flow. result Established the inequality for outermost generalized apparent horizons.
In this note, we obtain existence results for complete Ricci-flat Kahler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial…
Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken-Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard [t=0]-timeslice of the Schwarzschild solution. In this paper, w…
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.
Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.
problem Curvature pinching on asymptotically flat manifolds.
method Ricci flow on asymptotically flat manifolds with integral curvature pinching.
result Ricci flow converges to flat Euclidean space for sufficiently pinched curvature.
We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliat…
We consider several geometric inequalities in general relativity involving mass, area, charge, and angular momentum for asymptotically hyperboloidal initial data. We show how to reduce each one to the known maximal (or time symmetric) case in the asymptotically flat setting, whenever a geometrically motivated system of…
Constructs extensions for Bartnik data to approach mass limits.
problem Finding mass limits for Bartnik data.
method Shi-Tam type metric construction and refined monotonicity.
result Mass of constructed extensions can be made arbitrarily close to half area radius.
New proof of Penrose inequality using potential theory.
problem Proving the Riemannian Penrose inequality for black holes.
method Establishing a monotonicity formula for the p-capacitary potential.
result A new proof of the Penrose inequality for black holes.
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.
Proves spacetime positive mass theorem with corners.
problem Proving a positive mass theorem for spacetime with corners.
method Deformation theorem with corner conditions, asymptotically flat initial data.
result Exterior end satisfies E≥∣P∣ in every dimension n≥3. Flow of spacelike hypersurfaces converges to flat slice in asymptotically flat spacetimes.
problem Long-time behavior of mean curvature flow in asymptotically flat spacetimes.
method Analysis of mean curvature flow in Lorentzian product manifolds.
result Mean curvature flow converges uniformly to a flat slice as time goes to infinity.
The paper studies global Yamabe flow on AF manifolds, preserving ADM mass.
problem Existence and behavior of Yamabe flow on AF manifolds.
method New local existence theorem and maximum principle for parabolic equations.
result Global existence of Yamabe flow on AF manifolds with non-negative scalar curvature.
New method creates vacuum data at minimal and borderline decay thresholds.
problem Creating vacuum initial data at specific decay thresholds.
method Conical solution-operator method applied to vacuum asymptotically flat initial data.
result Demonstrates global and exterior stability of Minkowski spacetime.
In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to th…
Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of som…
Extends initial data to asymptotically flat solutions in general relativity.
problem Solving constraint equations for asymptotically flat solutions in general relativity.
method New method for solving prescribed divergence equation and geometric variant of conformal method.
result Global solution extending initial data to asymptotically flat solutions.
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.
Proves Penrose inequality for specific asymptotically flat manifolds.
problem Proving Penrose inequality for certain types of manifolds.
method Developed a new approximation scheme for a flow and established monotonicity of a free boundary Hawking mass.
result Proved the Riemannian Penrose inequality for specified manifolds.
Penrose conjecture proven for specific initial data sets.
problem Proving Penrose conjecture for certain types of initial data sets.
method Used σ-inverse mean curvature flow and a monotonicity formula.
result Penrose conjecture established for 2-convex initial data sets.
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
Researchers prove a Penrose inequality for spacetime with specific conditions.
problem Establishing mass lower bounds for spacetime with specific asymptotic conditions.
method Combining harmonic level set approach, Jang equation, and stability techniques.
result Proof of Penrose inequality with universal constant and minimal area requirement.
Constructs initial data for multiple black holes with specified ADM parameters.
problem Forming multiple black holes with specific ADM parameters.
method Smooth, asymptotically flat vacuum initial data with prescribed ADM energy, momentum, and angular momentum.
result Maximal development of data results in spacetimes containing multiple black holes.
Rigidity theorem shows massless hyperboloidal data embeds into Minkowski space.
problem Characterizing massless initial data sets in General Relativity.
method Precise decay estimates for spinors on harmonic level sets.
result Asymptotically hyperboloidal IDS with zero mass embed isometrically into Minkowski space.