On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.
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We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we …
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.
Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed Cheeger--Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. I…
In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compa…
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
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 consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.
Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.
problem Bounding total geodesic curvature in a hyperbolic setting.
method Derives an upper bound for total geodesic curvature in terms of boundary data.
result Upper bound for total geodesic curvature expressed solely in terms of boundary data.
Stability of positive mass theorem for hyperbolic graphs proven.
problem Proving stability of positive mass theorem for asymptotically hyperbolic graphs.
method Adapting ideas from previous work on asymptotically flat graphs to hyperbolic graphs.
result Stability of positive mass theorem for a class of n-dimensional asymptotically hyperbolic graphs.
Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.
problem Proving Gromov's conjecture on total mean curvature of fill-ins.
method Surgery to reduce to fill-ins of spheres, positive mass theorems, and quantitative surgery process.
result Proves Gromov's conjecture on total mean curvature in various cases.
Stability of positive mass theorem for hyperbolic manifolds studied.
problem Stability of the positive mass theorem for asymptotically hyperbolic manifolds.
method Adapted intrinsic flat distance approach to show stability for a class of manifolds.
result Stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds.
We prove that extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric, black holes. As a consequence…
Proof of mass positivity for hyperbolic space near Euclidean space.
problem Positivity of mass in asymptotically hyperbolic manifolds.
method Elementary proof generalizing Bartnik's Euclidean case.
result Positivity of hyperbolic mass near hyperbolic space.
Initial data with zero mass must be in pp-wave spacetimes.
problem Proving initial data with zero mass must be in pp-wave spacetimes.
method Spinorial methods combined with spacetime harmonic functions.
result Initial data with zero mass must be contained in pp-wave spacetimes.
Estimates Bartnik mass for metrics with nonnegative Gauss curvature.
problem Estimating Bartnik mass for specific metric configurations.
method Using area, total mean curvature, and a metric roundness measure.
result Estimate approaches sharp value for round spheres.
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…
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minko…
Study on pseudo-Einstein 3-manifolds for a specific inequality, introducing Robin mass.
problem Existence of contact structures on pseudo-Einstein CR manifolds.
method Introduced Robin mass and used it to study the variation of total mass under conformal change.
result Existence of a minimizer for total mass yielding the classical LHLS inequality.
Sharp mass bounds for ALE and ALF toric 4-manifolds.
problem Establishing lower bounds for the mass of ALE and ALF toric 4-manifolds.
method Using gravitational instantons and conical angle defects, the mass is bounded below by a sum of the mass of the corresponding instanton and an expression determined by conical angle defects.
result The mass of an ALE or ALF toric 4-manifold is not less than the mass of the corresponding gravitational instanton.
Novel concentration inequalities are obtained for the missing mass, i.e. the total probability mass of the outcomes not observed in the sample. We derive distribution-free deviation bounds with sublinear exponents in deviation size for missing mass and improve the results of Berend and Kontorovich (2013) and Yari Saeed…
We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the …
We show that extreme Myers-Perry initial data realize the unique absolute minimum of the total mass in a physically relevant (Brill) class of maximal, asymptotically flat, bi-axisymmetric initial data for the Einstein equations with fixed angular momenta. As a consequence, we prove the relevant mass-angular momentum in…
We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…
New geometric inequality for mass from immersed submanifolds.
problem Mass from immersed submanifolds in Euclidean space.
method Expressed mass as a linear combination of mean curvatures.
result Geometric inequality for positive mass theorem.
Paper proposes MMC to avoid high-density bias in clustering.
problem High-density bias in density-based clustering.
method Introduces mass distribution as a better foundation for clustering, proposing mass-maximization clustering (MMC).
result MMC avoids high-density bias and discovers clusters of arbitrary shapes, sizes, and densities.
Study calculates mass for certain hyperbolic spaces with noncompact boundaries.
problem Calculating mass for specific types of hyperbolic spaces.
method Defined a mass-type invariant for asymptotically hyperbolic manifolds with noncompact boundaries.
result Proved a sharp positive mass inequality under suitable conditions.
Positive mass theorem for tori with scalar curvature bounds.
problem Proving positivity of static quasi-local mass for tori.
method Generalization of Shi-Tam result to 2-tori with specific curvature and scalar curvature bounds.
result Total weighted mean curvature of 2-tori is not greater than that of an isometric embedding into the Kottler manifold.
We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group…
Paper proves stronger Penrose inequality with matter density.
problem Proves Penrose inequality with nonnegative matter density.
method Uses conformal flow and harmonic level set techniques.
result Total mass is at least black hole mass plus matter density contribution.
In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…
The paper proves a new theorem linking mass and electric charge for certain types of manifolds.
problem Proving a new positive mass theorem for manifolds with charge.
method Using conformal relations and scalar curvature, the authors derive a new theorem.
result The sum of mass is not less than the modulus of total electric charge under certain conditions.
Proves a positive mass theorem for static causal fermion systems.
problem Defining mass for complex spacetimes without regularity assumptions.
method Surface layer integrals comparing asymptotically flat and vacuum spacetimes.
result Proves a positive mass theorem for static causal fermion systems.
In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient co…
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvaturemath.DG
Paper studies minimal hypersurfaces and their impact on compact manifolds with nonnegative scalar curvature.
problem Analyzing the boundary behavior of compact manifolds with nonnegative scalar curvature.
method Examines the effect of minimal hypersurfaces on the boundary of compact manifolds.
result Establishes an inequality relating mass, area of minimal hypersurfaces, and weighted total mean curvatures.
We extend Sobolev transport to unbalanced measures on graphs.
problem Optimal transport struggles with measures of different total mass and high computational complexity.
method We propose a scalable unbalanced Sobolev transport (UST) for measures on graphs.
result UST admits a closed-form formula for fast computation and is negative definite.
Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative sca…
New mass-type invariants for cosmological space-times.
problem Characterize de Sitter solutions in space-times with a cosmological constant.
method Introduce new mass-type invariants and prove positive mass theorems.
result 1-harmonic Mass provides new characterizations and inequalities.
The study examines the behavior of monopoles as their mass increases and finds that they abelianize near singular points.
problem The behavior of monopoles as their mass increases and the formation of singular points.
method Analysis of mass-renormalized energy measures and convergence of fields to a reducible monopole.
result The mass-renormalized energy measures concentrate at singular points, and the fields abelianize near these points.
New foliation method for isolated systems in General Relativity.
problem Defining a robust center of mass for isolated systems in GR.
method Foliation by constant spacetime mean curvature (STCMC) 2-spheres.
result Unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set.
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative…
Mass inequality for black ring spacetimes, showing extreme solutions.
problem Establishing mass-angular momentum relation for black ring spacetimes.
method Mathematical derivation of mass-angular momentum inequality.
result The inequality is saturated for extreme Pomeransky-Sen'kov black ring solutions.
Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension math.DG
The study proves the regularity of inverse mean curvature flow in specific geometric settings.
problem Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds.
method Utilizing the behavior of Hawking masses, the study shows star-shaped slices after a long time.
result The weak solution of inverse mean curvature flow becomes regular over time.
Given a transportation cost , optimal maps minimize the total cost of moving masses from to . We find a pseudo-metric and a calibration form on such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like current…
Researchers extend Bartnik mass concept to hyperbolic spacetimes.
problem Quantifying quasi-local mass in asymptotically hyperbolic spacetimes.
method Constructing asymptotically hyperbolic extensions and controlling total mass.
result Established bounds for Bartnik mass in hyperbolic spacetimes.
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.
A simple model for unbalanced optimal transport captures key features.
problem Capturing the main features of unbalanced optimal transport.
method Introducing a metric on the conical extension of diffeomorphisms and studying its properties.
result Total mass evolves with constant acceleration along geodesics.