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.
The positive energy theorem is proven for certain spacetimes with irregular curvature.
problem Proving the positive energy theorem for spacetimes with irregular curvature.
method Weak asymptotically anti-de Sitter initial data sets with distributional curvature under weak dominant energy condition.
result Positive energy theorem established for weakly irregular spacetimes.
Proves energy positivity for certain hyperbolic manifolds.
problem Establishing energy positivity for asymptotically hyperbolic manifolds.
method Proved deformation-of-mass-aspect theorems and positivity results.
result Positivity of energy for a class of asymptotically hyperbolic manifolds in dimensions 4 to 7.
Paper proves positivity of energy function on Riemannian manifolds.
problem Investigating positivity of energy function on Riemannian manifolds.
method Using energy function to prove positivity of initial energy.
result Simple method to obtain growth of eigen-solutions.
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.
The hyperbolic positive energy theorem links causal properties to energy-momentum vectors in asymptotically hyperbolic spaces.
problem Establishing the causal-future-directed character of energy-momentum vectors in hyperbolic spaces.
method Analyzing n-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, focusing on the dominant energy condition. result The causal-future-directed character of the energy-momentum vector can be traced back to that of asymptotically Euclidean initial data sets.
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 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.
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.
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.
Study proves positive mass-energy theorems for specific spacetime manifolds.
problem Analyzing 3+1 rotationally symmetric Lorentzian Einstein spacetimes.
method Reduced equations to 2+1 Einstein equations coupled to wave maps, proving positive mass-energy theorems.
result Various explicit positive mass-energy theorems proved without smallness assumptions.
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.
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.
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…
K-energy is not strictly convex on certain complex manifolds, but under specific conditions, it is.
problem The strict convexity of Mabuchi's K-energy on geodesically complete spaces of bounded positive forms.
method Simple toric example and further assumptions on toric manifolds.
result Strict convexity holds in the toric case under certain conditions, leading to a uniqueness result.
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.
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. The study finds energy gaps for Yang-Mills fields on Kähler surfaces.
problem Finding energy gaps for Yang-Mills fields on Kähler surfaces.
method Proving an L2 energy gap result for Yang-Mills connections on Kähler surfaces with positive scalar curvature. result Proves energy gap results for Yang-Mills fields on Kähler surfaces.
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.
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 introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
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 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.
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: …
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.
Researchers construct a gauge-invariant energy functional for axially symmetric perturbations around Kerr black holes.
problem Understanding energy of axially symmetric perturbations around Kerr black holes.
method Hamiltonian dimensional reduction to a 2+1 Einstein-wave map system, constructing a positive-definite, gauge-invariant energy functional. result The energy functional serves as a Hamiltonian for the constrained evolution of linear perturbations.
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.