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.
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.
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.
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.
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 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 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.
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.
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.
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 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.
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. 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 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.
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.
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].
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…
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 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.
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…
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.
Surveying mass in 2D hyperbolic geometry, overcoming challenges via minimisation.
problem Defining mass in 2D hyperbolic geometry.
method Minimisation using positive energy theorem and gluing theorems.
result Construction of novel initial data sets with controlled mass.
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.
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 rigidity theorems for AE Q-singular spaces.
problem Analyzing Q-curvature on AE manifolds. method Introducing a fourth order energy and rewriting it in terms of a fourth order Ricci-like tensor.
result Yamabe positive J-flat AE manifolds are isometric to Euclidean space. 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.
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.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
New theorem for spacetime mass in noncompact regions.
problem Mass in noncompact spacetime regions.
method Developed mass type invariant and boundary conditions; proof based on spinors.
result Proved positive mass theorem for noncompact boundaries.
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.
Simplified proof of cosmic singularity theorem using new mathematical techniques.
problem Proving cosmic singularity in expanding spacetimes with positive cosmological constant.
method Unified approach using the positive resolution of the virtual positive first Betti number conjecture.
result The theorem holds without the need for a spherical Cauchy surface.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
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.
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.
We show that the causal-future-directed character of the energy-momentum vector of n-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, n≥3, can be traced back to that of asymptotically Euclidean general-relativistic initial data sets satisfying the dominant energy cond…
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM energy-momentum vector (E,P) of the initial data satisfies the inequality E≥∣P∣. Previously, this theorem was proven…
Positive mass theorem and Yamabe equation on CR manifolds
problem Positive mass theorem and Yamabe equation on CR manifolds
method Positive mass theorem and Yamabe equation on CR manifolds
result Positive mass theorem in 3-dimensional CR geometry
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.
In their proof of the positive energy theorem, Schoen and Yau showed that every asymptotically flat spacelike hypersurface M of a Lorentzian manifold which is flat along M can be isometrically imbedded with its given second fundamental form into Minkowski spacetime as the graph of a function from R^n to R; in particula…
Study on smoothness of 4D Willmore-type hypersurfaces.
problem Investigating smoothness of critical points of a 4D Willmore-type energy.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the energy are smooth.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, U(1)2-invariant initial data sets on simply connected four dimensional manifolds Σ. Moreover, we extend the local mass angular momenta inequality result Ref [1] for U(1)2 invariant black holes to the case with nonzero stre…
In this paper, we introduce a new energy density function Y on the projective bundle P(TM)M for a smooth map f:(M,h)(N,g) between Riemannian manifolds Y=gijfαifβj∑hγδWγWδWαWβ. We get new Hessian estimates to this energy density and obtain various new…