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.
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.
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.
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.
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.
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
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.
In [5], a rigidity result was obtained for outermost marginally outer trapped surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This allowed one to treat the "borderline case" in the author's work with R. Schoen concerning the topology of higher dimensional black holes [8]. The proof of this rigi…
Paper proves rigidity for spin bands with specific conditions.
problem Proving rigidity for initial data sets on spin bands.
method Using Dirac operator techniques and lightlike imaginary W-Killing spinors. result Obtains slight generalizations of known rigidity results.
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.
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.
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.
The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.
problem Geometric constraints near surfaces with equality in area-charge inequalities.
method Investigation of equality in area-charge inequalities for spherical minimal surfaces and MOTS within the Einstein-Maxwell equations framework.
result Equality in area-charge inequalities imposes rigid geometric structures, including normal electric and magnetic fields and isometric Riemannian products.
The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
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.
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
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.
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.
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.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
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.
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.
Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.
problem Proving rigidity of convex polytopes in hyperbolic space.
method Spinor techniques and recent smoothing constructions of Brendle-Wang.
result Scalar curvature rigidity for parabolic convex polytopes in hyperbolic space.
We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on Rn , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, w…
We describe a proof of M.T. Anderson's result on the rigidity of complete stationary initial data for the Einstein vacuum equations in spacetime dimension 3 + 1, under an extra assumption on the norm of the stationary Killing vector field. The argument only involves basic comparison geometry along with some Bochner-Wei…
We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers-Perry black hole initial data are the unique minimizers. In addition, we establish a rigi…
In arXiv:0905.2622v1 and arXiv:0910.4785v1, Bray and Khuri outlined an approach to prove the Penrose inequality for general initial data sets of the Einstein equations. In this paper we extend this approach so that it may be applied to a charged version of the Penrose inequality. Moreover, assuming that the initial dat…
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 4π/c, where c>0 is a lower bound on a natural energy-momentum term. We then consider th…
The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper we address the question of what happens when the mass is …
A lower bound for the ADM mass is established in terms of angular momentum, charge, and horizon area in the context of maximal, axisymmetric initial data for the Einstein-Maxwell equations which satisfy the weak energy condition. If, on the horizon, the given data agree to a certain extent with the associated model Ker…
We present several rigidity results for Riemannian manifolds (Mn,g) with scalar curvature S≥−n(n−1) (or S≥0), and having compact boundary N satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of…
The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.
problem The rigidity and positivity of the Hawking energy on specific surfaces in general relativity.
method Evaluation of the Hawking energy on area-constrained critical surfaces under the dominant energy condition.
result The Hawking energy is nonnegative and rigid on area-constrained surfaces, including charged and cosmological constant variants.
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…
Proves rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.
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.
Paper proves rigidity of de-Sitter tori with conical singularities.
problem Global rigidity of de-Sitter tori with singularities.
method Introduced constant curvature Lorentzian surfaces with conical singularities and proved rigidity via topological dynamics.
result De-Sitter tori with a single singularity are determined by their lightlike bi-foliation.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention and a cohomogeneity-one analytic system, proving local existence for nonconstant mean curvature and dilation.
result Local existence and rigidity results for biharmonic conformal immersions into anti-de Sitter space.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention, expressing biharmonic equation in terms of induced metric and curvature, deriving cohomogeneity-one analytic system, and solving scalar third-order ODE.
result Local existence and rigidity of biharmonic conformal immersions with nonconstant dilation.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m≥3 starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe…
In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positi…
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
Stability and rigidity of Ricci-flat ALE manifolds proven.
problem Stability and rigidity of Ricci-flat ALE manifolds.
method Proved stability and rigidity of ALE manifolds with a parallel spinor under Ricci flow, given initial metrics close in Lp∩L∞. result Strong decay rates prove positive scalar curvature rigidity in Lp for each p∈[1,n−2n). 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.
We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as u…
In this paper we study some global properties of static potentials on asymptotically flat 3-manifolds (M,g) in the nonvacuum setting. Heuristically, a static potential f represents the (signed) length along M of an irrotational timelike Killing vector field, which can degenerate on surfaces corresponding to the…
We affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight. Namely, if an asymptotically flat initial data set satisfies the dominant energy condition and has E=∣P∣, then E=∣P∣=0, where (E,P) is the ADM energy-momentum vector. The dimensional restriction can be removed…
The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.
problem Proving a positive mass theorem for non-compact static domains in hyperbolic space.
method Formulating and proving a positive mass theorem under natural dominant energy conditions, using elliptic boundary conditions on spinors.
result Retrieve a sharper version of a recent result by Souam about the rigidity of non-compact static domains.