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.
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.
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 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.
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.
We present an elementary argument that one can shield linearised gravitational fields using linearised gravitational fields. This is done by using third-order potentials for the metric, which avoids the need to solve singular equations in shielding or gluing constructions for the linearised metric.
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.
Model analyzes debt recycling strategies under various fiscal regimes and jurisdictions.
problem Understanding debt recycling dynamics and their impact on repayment times and equity growth.
method Developed a calibrated model incorporating mortgage interest rates, borrowing costs, and tax shields.
result Introducing positive interest rates without tax shields contracts success regions and lengthens repayment times, but tax shields partially reverse these effects.
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. Shielded LMC samples from non-convex spaces with repulsive drift.
problem Sampling from non-convex spaces with convex holes.
method Combining adaptive temperature and repulsive drift.
result Advantages over unconstrained sampling in constrained spaces.
We evaluate the applicability of the generic Vickrey-Clarke-Groves (VCG) mechanism as an antimonopoly measure against a profit-maximizing producer with market power operating a portfolio of generating units at the centralized two-settlement energy market. The producer may indicate in its bid not only the altered cost f…
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 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.
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.
We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.
Generalizes nonnegativity result for Brown-York mass using noncompact fill-ins.
problem Proving nonnegativity of Brown-York mass for noncompact fill-ins.
method Extending positive mass theorem to noncompact and shielded fill-ins.
result Proves nonnegativity of mean curvature for noncompact NNSC fill-ins.
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.
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…
In this appraisal paper, we evaluate the efficacy of SHIELD, a compression-based defense framework for countering adversarial attacks on image classification models, which was published at KDD 2018. Here, we consider alternative threat models not studied in the original work, where we assume that an adaptive adversary …
New method provides scalable safety guarantees for RL agents.
problem Safe reinforcement learning in real-life scenarios.
method State-augmentation and shield design for probabilistic avoidance.
result Strict formal safety guarantees for RL agents, scalable and practical.
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.
Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.
problem Overcoming singularities in the Schoen-Yau proof for arbitrary dimensions.
method Inductive scheme combining shielding principle, conformal blow-up, and Cheeger-Naber bound.
result Proof of positive mass theorem in arbitrary dimensions.
Reinforcement learning is a promising approach to synthesizing policies for challenging robotics tasks. A key problem is how to ensure safety of the learned policy---e.g., that a walking robot does not fall over or that an autonomous car does not run into an obstacle. We focus on the setting where the dynamics are know…
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.
We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our constructio…
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.
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…
Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G…
The Bethe free energy approximation is reliable when convex on a submanifold, the 'Bethe box'.
problem Accuracy of the Bethe free energy approximation in probabilistic inference.
method Analysis of convexity and verification conditions based on the Bethe Hessian matrix.
result The Bethe approximation is mostly accurate if it is convex on a submanifold, the 'Bethe box'.
Minimal surfaces connect to horizons and electrostatic systems.
problem Connecting minimal surfaces to horizons and electrostatic systems.
method One-parameter min-max problem for area functional, inequality relating area and charge.
result Minimal surfaces of index one are related to unstable horizons in electrostatic systems.
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…
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…
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 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.
The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the…
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.
Graph neural networks are explained through energy gradient flow and framelet decomposition.
problem Understanding and improving graph neural networks.
method Viewing framelet-based models as gradient flows of energy, proposing a generalized energy via framelet decomposition.
result The proposed model leads to more flexible dynamics, enhancing graph neural networks.
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.
Improves diversity of text-to-image models without sacrificing FID.
problem Lack of diversity and tendency to recreate training set images.
method Adds sparse repellency terms to diffusion SDE to guide trajectories away from a reference set.
result Improves diversity of diffusion models with minimal impact on FID.
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…
Localized curvature bounds ensure harmonic maps are constant.
problem Ensuring harmonic maps are constant under localized curvature constraints.
method Localized Bochner-type rigidity theorem for harmonic maps with image-dependent curvature bounds.
result Harmonic maps are constant if minimal Ricci curvature dominates image-dependent curvature bounds.
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.
In this survey article we review several results on the curvature of semi-Riemannian metrics which are motivated by the positive mass theorem. The main themes are estimates of the Riemann tensor of an asymptotically flat manifold and the construction of Lorentzian metrics which satisfy the dominant energy condition.