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.
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.
The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik…
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
Proves properties of free boundary stable MOTS in spacetimes.
problem Topology of black hole spacetimes in manifolds with boundary.
method Initial data version of Hawking's theorem, foliation by MOTS, vanishing null second fundamental form.
result Compact free boundary stable MOTS in initial data sets with boundary are of positive Yamabe type.
Building upon the work of Brendle, Marques and Neves on the construction of counterexamples to Min-Oo's conjecture, we exhibit deformations of the de Sitter-Schwarzschild space of dimension n≥3 satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons…
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.
A purely algebraic construction of super-energy tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely …
We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condit…
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.
Extends results on marginally outer trapped surfaces to general null expansion.
problem Analyzing geometry and topology of expanding horizons.
method Introduces g-stability and proves conditions for positive Yamabe type and scalar curvature. result Initial data sets with compact boundary of positive null expansion have positive mass.
Study shows inflation in 3+1D cosmologies with bounded scalar potential and specific symmetry.
problem Understanding inflation in 3+1D cosmologies with specific constraints.
method Mean curvature flow and asymptotic analysis of metric variations, stress-energy tensor, and inflaton field dynamics.
result Inflation occurs in 3+1D cosmologies with specific constraints, demonstrating it is possible with inhomogeneous initial conditions.
The study shows how energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces.
problem Understanding energy density and topological invariants in n-Fuchsian fibers of Higgs bundles. method Establishing an algebraic inequality generalizing a GIT theorem to prove energy density domination.
result Energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces. 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.
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. Study of Dirac-Witten operator on Lorentzian manifolds under dominant energy condition.
problem Detecting non-trivial homotopy groups in spaces of initial data under strict dominant energy condition.
method Use index theory and Lorentzian Hitchin's α-invariant to analyze Dirac-Witten operator.
result Kernel of Dirac-Witten operator is non-trivial only if fundamental group is virtually solvable of derived length at most 2.
We consider branes N in a Schwarzschild-AdS(n+2) bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential V, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field …
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.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
Proves non-existence of certain metrics on specific manifolds.
problem Existence of metrics with nonnegative scalar curvature.
method Connected sum technique and non-compact domination method.
result Connected sum of aspherical manifolds with non-compact manifolds does not admit metrics with nonnegative scalar curvature.
Paper proves Penrose inequality with a weaker late-time condition.
problem Penrose's inequality under the black hole final state conjecture.
method Developed a new late-time condition called quasi final state hypothesis and proved the inequality.
result Proved the spacetime Penrose inequality under the quasi final state hypothesis.
We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.
Smooth dec initial data sets may not extend to smooth spacetimes.
problem Whether every dec initial data set can be extended to a smooth spacetime.
method Examined the converse of the dominant energy condition for initial data sets and spacelike hypersurfaces.
result Not all dec initial data sets can be extended to smooth spacetimes.
Extends K-energy to complexified Kähler classes for scalar curvature study.
problem Scalar curvature equation with B-field on complexified Kähler classes.
method Extended K-energy functional, convex along geodesics.
result Uniqueness of solutions in some cases.
Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.
problem Understanding the behavior and stability of charged scalar fields on near-extremal Reissner-Nordström spacetimes.
method Global integrated energy decay and boundedness estimates for solutions to the charged scalar field equation.
result Established global, weighted integrated energy decay and boundedness estimates for solutions on (near-)extremal Reissner-Nordström(--de Sitter) spacetimes.
Study harmonic maps and anti-de Sitter 3-manifolds.
problem Existence and uniqueness of harmonic maps in infinite energy settings.
method Generalization of Corlette's result to infinite energy, analysis of asymptotic behavior, use of CAT(-1) Hadamard manifolds.
result Existence of new anti-de Sitter 3-manifolds.
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.
Proves symmetries of extremal horizons in spacetimes.
problem Proving symmetries of extremal horizons in arbitrary dimensions.
method Analyzes Killing vector fields and near-horizon geometry.
result Enhanced isometry groups and shifted Aretakis instability.
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…
We show that many Lorentzian manifolds of dimension >2 do not admit a spacelike codimension-one foliation, and that almost every manifold of dimension >2 which admits a Lorentzian metric at all admits one which satisfies the dominant energy condition and the timelike convergence condition. These two seemingly unrelated…
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 dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its c…
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
Energy Matching unifies flow matching and energy-based models for generative modeling.
problem Inability of flow-based models to integrate partial observations and priors.
method Energy Matching framework that integrates flow matching and energy-based models.
result Substantially outperforms existing EBMs on CIFAR-10 and ImageNet generation.
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
Based on Donaldson's method, we prove that, for an integral Kahler class, when there is a Kahler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.
Study of Bondi-Sachs formalism for massless scalar field with zero cosmological constant.
problem Analyzing the Bondi-Sachs formalism for Einstein's massless scalar field equations.
method Asymptotic expansions and peeling property for Bondi-Sachs metrics and scalar fields.
result Positivity of Bondi energy-momentum under specific conditions.
Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.
problem Analyzing late-time behavior and stability of charged scalar fields on black hole backgrounds.
method Purely physical-space based methods, energy estimates, inverse-power laws.
result First pointwise decay estimates for charged scalar fields on black hole backgrounds.
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.
Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.
problem Verifying scalar-flatness for critical metrics in specific dimensions.
method Analyzing complete Riemannian manifolds with critical metrics of the L2-scalar curvature functional. result The conjecture that all complete noncompact critical metrics with finite energy are scalar-flat is confirmed for dimensions 5 to 9.
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.
In this note we give a simplified proof of a recent result of X.X. Chen, which together with work of G. Szekelyhidi implies that on a sufficiently small deformation of a polarized constant scalar curvature Kahler manifold the K-energy has a lower bound.
Impact of projective curvature tensor in f(R,G), f(R,T) and f(R,Lm)-gravitygr-qc The study characterizes spacetime and modified gravity models using projective curvature tensor.
problem Characterizing spacetime and modified gravity models with projective curvature tensor.
method Analyzing $f\left(R,G
ight)$, $f\left(R,T
ight)$, and $f\left(R,L_{m}
ight)$-gravity models.
result Projectively flat perfect fluid spacetimes represent dark energy era and are locally isometric to Minkowski or de-Sitter spacetimes.
Formula derived for ALH manifolds, showing existence of specific 3D manifolds.
problem Energy calculation and existence of specific 3D conformally compact ALH manifolds.
method Derivation of energy formula and application to manifold existence.
result Existence of 3D conformally compact ALH manifolds with specific properties.
New method ranks multivariate distributions in SMOOP using q-dominance.
problem Lack of reliable methods to rank multivariate distributions in SMOOP.
method Introduces center-outward q-dominance and develops empirical test procedures.
result Proves q-dominance implies FSD and establishes a sample size threshold.
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.
In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…
We establish a Penrose-Like Inequality for general (not necessarily time symmetric) initial data sets of the Einstein equations which satisfy the dominant energy condition. More precisely, it is shown that the ADM energy is bounded below by an expression which is proportional to the square root of the area of the outer…