Abstract reviews hyperbolic positive energy theorems.
problem Analyzing positive energy theorems for hyperbolic spaces.
method Review of existing literature on asymptotically hyperbolic manifolds.
result Summarizes positive energy theorems for hyperbolic spaces.
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 a higher-order positive energy theorem for stationary solutions in fourth-order gravity.
problem Proving a positive energy theorem for fourth-order gravitational theories.
method Analyzes geometric analysis intersections and links to Q-curvature. result Establishes a positive energy theorem for stationary solutions in fourth-order gravity, similar to the classical ADM theorem.
Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfie…
New proof shows spacetime energy is always positive in higher dimensions.
problem Proving spacetime positive energy in arbitrary dimensions.
method Combines Schoen-Yau, Eichmair, Jang equation, shielding principle.
result Spacetime positive energy theorem proven in arbitrary dimensions.
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 positive energy conjecture for a specific metric class.
problem Proving the positive energy conjecture for a class of AHM metrics.
method Analyzes asymptotically Horowitz-Myers (AHM) metrics on R2imesTn−2. result Generalizes previous results on positive energy conjecture.
We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.
We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.
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.
In this paper, energy function is used to investigate the eigen-solutions of −Δu+Vu=λu on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.
We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.
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.
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
We give a short review of recent progress on the positive energy theorem in general relativity, especially for spacetimes with nonzero cosmological constant.
We establish a type of positive energy theorem for asymptotically anti-de Sitter Einstein-Maxwell initial data sets by using Witten's spinoral techniques.
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
We extend the positive mass theorem proved previously by the author to the Lorentzian setting. This includes the original higher dimensional positive energy theorem whose spinor proof was given by Witten in dimension four and by Xiao Zhang in dimension five.
This study proves energy bounds in specific AdS spacetimes.
problem Proving positive energy theorems in asymptotically locally AdS spacetimes.
method Derived positive energy theorem for spacetimes with compact, Einstein cross-sections.
result First complete proofs of BPS inequalities in AdS and locally AdS spacetimes.
New energy definition for expanding de Sitter spacetime with umbilic boundaries.
problem Defining energy for spacetimes with expanding de Sitter background and umbilic boundaries.
method Adapting Liu-Yau energy to a quasi-local setting in expanding de Sitter spacetime.
result Positivity of the defined energy for certain values of the cosmological constant.
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.
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 stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fa…
Researchers found a way to measure energy in black hole perturbations.
problem Lack of positive-definite and conserved energy in black hole stability.
method Dimensional reduction and construction of a positive-definite energy functional.
result Conserved Hamiltonian energy for axially symmetric perturbations of Kerr black holes.
New proof shows equality in spacetime mass theorem.
problem Proving the equality case of spacetime positive mass theorem.
method Uses a new approach requiring only E≥∣P∣ for near initial data sets. result Initial data sets with null ADM energy-momentum must embed into Minkowski space.
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. A generalized positive energy theorem for spaces with asymptotic SUSY compactification involving non-symmetric data is proved. This work is motivated by the work of Dai [D1][D2], Hertog-Horowitz-Maeda [HHM], and Zhang [Z].
The paper extends the spacetime positive mass theorem to multiple time dimensions.
problem Proving the nonnegativity of mass in spacetimes with multiple time dimensions.
method Generalizing the spacetime positive mass theorem to include multiple time dimensions and showing mass nonnegativity through energy inequalities.
result Equality in the energy inequality implies a foliation by flat submanifolds.
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.
We prove positivity of energy for a class of asymptotically locally hyperbolic manifolds in dimensions 4≤n≤7. The result is established by first proving deformation-of-mass-aspect theorems in dimensions n≥4. Our positivity results extend to the case n=3 when more stringent conditions are imposed.
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.
We prove the following stronger verson of the positivity of quasi-local mass stated in gr-qc/0303019: the quasi-local energy (mass) of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the space…
CLuP achieves near optimal ground state energies for positive and negative Hopfield models.
problem Finding near optimal ground state energies for positive and negative Hopfield models.
method Controlled Loosening-up (CLuP) algorithm with fully lifted random duality theory (fl RDT).
result Achieves ground state free energies of 1.77 and 0.33 for positive and negative Hopfield models respectively. The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.
This paper proves a positive energy-momentum theorem for oriented Riemannian 3-manifolds that are asymptotic to a standard hyperbolic slice in anti de Sitter space-time. Analogously to the original Witten's proof in the asymptotically flat case, this result relies on spinorial methods. We also give a rigidity theorem: …
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…
Paper proves existence of minimum energy solutions in 5D contact spin manifolds.
problem Finding minimum energy solutions for CR Yamabe equation in 5D contact spin manifolds.
method Spinorial approach based on a positive mass theorem.
result Existence of minimum energy solutions in 5D contact spin manifolds.
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
Classifies positive solutions to critical p-Laplace equation.
problem Classifying positive solutions to a specific type of partial differential equation.
method Analyzes solutions on \(\mathbb{R}^n\) with various energy growth conditions and infinity behavior.
result Provides classification under different conditions, including rigidity in some cases.
We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension n=3 to dimensions 3≤n<8. This requires us to address several technical difficulties that are not present when n=3. The regularity and decay assumptions for the initial data sets to which our argume…
New insights into Bartnik mass from improvability of dominant energy scalar.
problem Characterizing Bartnik mass minimizing initial data sets.
method Introducing improvability concept, proving non-improvability consequences, and analyzing pp-wave counterexamples.
result Bartnik mass minimizing initial data sets are characterized, advancing conjectures.
The study examines representations of compactly supported diffeomorphisms with a positive energy condition.
problem Analyzing projective unitary representations of compactly supported diffeomorphisms with a generalized positive energy condition.
method Investigates continuous second Lie algebra cohomology and uses it as an intermediate step to show that such representations are trivial on the identity component.
result Any such representation is trivial on the identity component of the group of compactly supported diffeomorphisms if the manifold is connected and has dimension greater than 1.
The study proves a new positive energy theorem for manifolds with specific curvature properties.
problem Proving a new positive energy theorem for manifolds with specific curvature properties.
method Establishing a systolic inequality relating boundary mean curvature to the systole of the boundary.
result Obtaining a new positive energy theorem, with equality for Horowitz-Myers metrics.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
Paper discusses quasilocal mass and fill-ins, proving positivity and exploring definitions.
problem Exploring and defining quasilocal mass and fill-ins in general relativity.
method Analyzes several proposals of quasilocal mass based on Hamiltonian formulation and proves positivity under certain conditions.
result Positivity of Wang-Yau energy under a more general condition.
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.