The paper connects -manifolds to metric dynamics.
problem Understanding the dynamics of metric spaces.
method Studied a Hamiltonian function on Riemannian metrics.
result Orbits correspond to -manifolds foliated by hypersurfaces.
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48 results for “ADM formalism”
The paper shows how to create scalar flat metrics with very large ADM mass.
problem Understanding the ADM mass of scalar flat Kähler ALE spaces.
method Blowing up points in ALE spaces to increase ADM mass.
result It is possible to produce scalar flat metrics with arbitrarily large ADM mass.
The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
problem Proving the X-positive mass theorem for all dimensions.
method Conformal reduction argument.
result The X-ADM mass is equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
New ADM mass definition for weakly regular manifolds.
problem Defining ADM mass for non-smooth manifolds.
method Proposed a new definition for metrics with local Sobolev regularity.
result Finite mass, invariance under coordinate changes, and agreement with smooth case.
When a spacetime takes Bondi radiating metric, and is vacuum and asymptotically flat at spatial infinity which ensures the positive mass theorem, we prove that the standard ADM energy-momentum is the past limit of the Bondi energy-momentum. We also derive a formula relating the ADM energy-momentum of any asymptotically…
Proves critical points of ADM mass correspond to specific initial data sets.
problem Finding initial data sets with fixed Bartnik boundary data.
method Proves existence of critical points on a Banach manifold.
result Critical points of ADM mass correspond to initial data sets with generalized Killing vector fields.
New relation found between ADM mass and generalized Komar energy for dynamical spacetimes.
problem Finding equality between ADM mass and Komar energy in dynamical spacetimes.
method Constructing a generalized Komar energy from the normal evolution vector and proving equality under specific conditions.
result Equality between ADM mass and generalized Komar energy for dynamical asymptotically-flat spacetimes.
In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on -dimensional, , asymp…
Study shows a mass quantity for metrics that agrees with ADM mass.
problem Understanding ADM mass for metrics and its behavior under Ricci-DeTurck flow.
method Developed a mass quantity and analyzed its behavior under Ricci-DeTurck flow.
result The mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow.
Two masses on surfaces with boundary converge to ADM mass.
problem Evaluating quasi-local masses on surfaces with boundaries.
method Hawking mass and Huisken's isoperimetric mass on surfaces with boundary, convergence to ADM mass.
result Convergence of Hawking and Huisken's masses to ADM mass.
Study bounds outer surfaces in small mass geometrostatic manifolds.
problem Bounding outer surfaces in geometrostatic manifolds with small ADM mass.
method Proving Intrinsic Flat Stability of the Positive Mass Theorem.
result Stability of the Positive Mass Theorem in geometrostatic manifolds with small ADM mass.
In a vacuum spacetime equips with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation ({\bf Condition D}), we establish the relation between the ADM total linear momentum and the Bondi momentum. The relation between the ADM total energy and the Bondi mass in t…
Constructs initial data for multiple black holes with specified ADM parameters.
problem Forming multiple black holes with specific ADM parameters.
method Smooth, asymptotically flat vacuum initial data with prescribed ADM energy, momentum, and angular momentum.
result Maximal development of data results in spacetimes containing multiple black holes.
New mass definition linked to ADM mass for general metrics.
problem Defining mass for metrics with low regularity.
method Using isocapacitary inequality to define total mass.
result Inequality between new mass and ADM mass proved.
We give some lower estimates of the ADM mass of an asymptotically flat (AF) Riemannian manifold without assuming that the scalar curvature of the manifold is nonnegative. Some sufficient conditions for an AF manifold to have nonnegative ADM mass are obtained. We also give some lower estimates of the Brown-York mass of …
Lower semicontinuity of ADM mass proven for a weaker convergence type.
problem Behavior of ADM mass under weak convergence types.
method Intrinsic flat convergence, smooth manifolds converging to local integral current spaces, Huisken's isoperimetric mass.
result Lower semicontinuity of ADM mass proven for convergence.
Predicts financial asset dependencies using spatiotemporal patterns.
problem Complex dependency structures in financial assets for risk mitigation.
method Proposes Asset Dependency Matrix (ADM) and Asset Dependency Neural Network (ADNN) with ConvLSTM for spatiotemporal asset dependency prediction.
result ADNN outperforms baselines in predicting asset dependencies and their applications.
Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.
problem Finding unique isoperimetric surfaces in flat 3-manifolds.
method Used 'fill-in' argument and sharp isoperimetric inequality.
result Each leaf of the canonical foliation is the unique isoperimetric surface.
The paper extends semicontinuity of ADM mass to dimensions 2-7.
problem The semicontinuity of ADM mass in asymptotically flat metrics.
method Using recent work on the Riemannian Penrose inequality, the paper extends semicontinuity from dimension 3 to 7.
result The semicontinuity of ADM mass is proven for dimensions 2-7.
New formula shows how causal vectors relate to mass-minimizing data.
problem Understanding mass-minimizing initial data sets and their geometry.
method Developed a new monotonicity formula for causal Killing vectors.
result Established strong maximum principles for the Lorentzian length.
The paper defines a new mass quantity for 3-manifolds and proves a positive mass theorem.
problem Proving the positive mass theorem for a new geometric quantity.
method Defining -ADM mass and using a monotonicity formula.
result Established a relative positive mass theorem for asymptotically flat 3-manifolds.
We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of …
We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…
Online optimization has emerged as powerful tool in large scale optimization. In this paper, we introduce efficient online algorithms based on the alternating directions method (ADM). We introduce a new proof technique for ADM in the batch setting, which yields the O(1/T) convergence rate of ADM and forms the basis of …
We consider an asymptotically flat Lorentzian manifold of dimension (1,3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when th…
Proves mass theorem for spacetimes with irregular curvature.
problem Proving mass theorem for spacetimes with irregular curvature.
method Introduced distributional curvature and generalized ADM momentum.
result Proved Lorentzian positive mass theorem.
Proves Green function rigidity for specific operators and obtains new ADM mass formula.
problem Proving Green function rigidity for specific operators and obtaining new ADM mass formula.
method Positive mass theorem and positive energy theorem for Paneitz operator.
result Obtained new formula for the ADM mass of asymptotically flat hypersurfaces.
We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the sa…
The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetr…
A definition of the mass aspect function for weakly regular asymptotically hyperbolic manifoldsmath.DG
Unified definition of mass aspect function for weakly regular hyperbolic manifolds.
problem Ambiguity in mass definition for asymptotically hyperbolic manifolds.
method Introduced an ADM-style mass aspect function for broad asymptotics and low regularity.
result Unified mass aspect function exhibits favorable covariance properties.
Simple proof for sphere mass calculation.
problem Computing the ADM mass of static sphere extensions.
method Uses mass formula for static asymptotically flat manifolds.
result Validated mass formula for small spheres.
We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy-momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig-Ó Murchadha-Regge-T…
The paper studies global Yamabe flow on AF manifolds, preserving ADM mass.
problem Existence and behavior of Yamabe flow on AF manifolds.
method New local existence theorem and maximum principle for parabolic equations.
result Global existence of Yamabe flow on AF manifolds with non-negative scalar curvature.
On the relation between ADM and Bondi energy-momenta III -- perturbed radiative spatial infinitygr-qc
In a vacuum spacetime equipped with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation ({\bf Condition D}), we establish the relation between the ADM total energy-momentum and the Bondi energy-momentum for perturbed radiative spatial infinity. The perturbatio…
Mass in relativity linked to polyhedra geometry.
problem Understanding ADM mass in general relativity.
method Relating ADM mass to the total mean curvature and defect of dihedral angles of Riemannian polyhedra.
result Expressed -dimensional mass as an integral of geometric quantities.
The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.
problem Generalizing spin geometry to weighted manifolds and defining a new mass.
method Investigates spectral properties of the weighted Dirac operator and defines a new mass.
result Defines a new mass for weighted asymptotically Euclidean manifolds and shows its monotonicity under Ricci flow.
Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
problem Defining a mass for asymptotically hyperbolic manifolds under weaker conditions.
method Volume-renormalized mass defined as a linear combination of ADM mass and renormalized volume.
result Volume-renormalized mass is well-defined and diffeomorphism invariant under weaker conditions.
The study proves stability of the positive mass theorem for Kähler manifolds.
problem Stability of the positive mass theorem for Kähler manifolds.
method Integral inequality and stability results for ADM mass on AE Kähler manifolds.
result Stability of the positive mass theorem for Kähler manifolds under certain conditions.
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
The paper proves a new inequality for 3-manifolds with noncompact boundaries.
problem Proving positivity of a convex combination of ADM masses on 3-manifolds with noncompact boundaries.
method Obtained an integral inequality for asymptotically linear harmonic functions.
result Positivity of a convex combination of ADM masses under a positivity condition on scalar curvatures and boundary mean curvatures.
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.
problem Estimating the mass of 3-manifolds with non-negative scalar curvature and minimal boundary.
method Derives monotone quantities for p-harmonic functions and applies them to derive a sharp mass-capacity estimate.
result Derives a sharp mass-capacity estimate relating the ADM mass of a 3-manifold to the p-capacity of its boundary.
The paper establishes preferred coordinates for AE 3-manifolds, improving ADM center of mass convergence.
problem Establishing preferred coordinates for asymptotically Euclidean 3-manifolds.
method Analyzing regularity of conformal compactifications via elliptic theory.
result Improves Sobolev regularity of conformally compactified AE 3-manifolds.
We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian characte…
Study of Brown--York mass for four-dimensional asymptotically flat manifolds.
problem Calculating mass for hypersurfaces in four-dimensional asymptotically flat manifolds.
method Intrinsic definition of mean curvature, expansion analysis for large uniformly convex hypersurfaces.
result Shape-dependent correction to ADM mass for nearly round surfaces vanishes under certain conditions.
The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
problem Proving a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
method Introducing a one-parameter family of functions that are monotone along the level-set flow of the potential, up to the optimal threshold.
result Proves a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.