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.
Proves positive mass theorem on conical manifolds with small angles.
problem Proving the positive mass theorem on conical manifolds with small cone angles.
method Analyzes conical manifolds with small cone angles, assuming spin structure and locally conformal flatness.
result Proves the positive mass theorem under specified conditions.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
Proves positive mass theorem for specific manifold types.
problem Positive mass theorem for manifolds with arbitrary ends.
method Proof for asymptotically flat and Euclidean manifolds.
result Validates positive mass theorem in new manifold types.
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.
Paper proves mass theorems for nonnegative scalar curvature metrics.
problem Proving mass theorems for metrics with nonnegative scalar curvature.
method New local inverse mean curvature flow with quantitative stability.
result Existence of isoperimetric sets in low regularity metrics.
Study proves positivity of quasi-local masses in general relativity using spinors.
problem Proving the positivity of quasi-local masses in general relativity.
method Using spinors and solving Dirac equation on compact Riemannian manifolds with boundary conditions.
result Gravitational mass bounded by a spacelike topological 2-sphere is non-negative, vanishing only in Minkowski space.
New positive mass theorems for ALH manifolds with toroidal ends.
problem Proving positive mass theorems for asymptotically locally hyperbolic manifolds.
method Utilizes properties of marginally outer trapped surfaces and a new technique involving μ-bubbles.
result Obtained new positive mass theorems for asymptotically locally hyperbolic manifolds without boundary.
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.
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.
Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.
problem Characterizing ALH manifolds with boundary and their mass.
method Scalar curvature deformation analysis and mass rigidity study.
result ALH manifolds that minimize mass integrals are characterized.
In this paper, we obtain a positivity result of a quasi-local mass integral as proposed by Shi and Tam in general dimensions. The main argument is based on the monotonicity of a mass integral in a foliation of quasi-spherical metrics and a positive mass type theorem which was proved by Wang and Yau in the three dimensi…
We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown--York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown--York mass of the boundary of some co…
The paper proves mass nonnegativity for certain asymptotically locally flat manifolds.
problem Proving mass nonnegativity for asymptotically locally flat manifolds.
method Using positive mass theorems and scalar curvature assumptions.
result Mass is nonnegative for specified asymptotically locally flat manifolds.
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.
Continuous metrics on R^3 with specific properties have non-negative harmonic mass.
problem Proving non-negativity of mass for continuous metrics.
method Defining harmonic mass and using properties of approximating smooth metrics.
result The harmonic mass of continuous metrics is non-negative.
The paper classifies and proves properties of ALE manifolds and orbifolds.
problem Classifying and understanding the structure of ALE manifolds and orbifolds.
method Proving structure theorems and using properties of conformal blow-up and compactification.
result Several classifications and properties of ALE manifolds and orbifolds are given in low dimensions.
Proves spacetime positive mass theorem in all dimensions.
problem Proving the spacetime positive mass theorem in arbitrary dimensions.
method Using Brendle--Wang's Riemannian positive mass theorem approach.
result Proves the spacetime positive mass theorem for all dimensions.
W. Simon proved a conformal positive mass theorem, which was used to prove uniqueness of black holes later. In this note, we will generalize Simon's conformal positive mass theorem in two directions. First we will consider spacetime version of conformal positive mass theorems on asymptotically flat initial data set. Ne…
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
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 X-ADM mass and using a monotonicity formula. result Established a relative positive mass theorem for asymptotically flat 3-manifolds.
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.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Proves positive mass theorem for 3-manifolds with a boundary.
problem Proving the positive mass theorem for specific 3-manifolds.
method Uses harmonic level set approach.
result Validates the positive mass theorem for new class of manifolds.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
Formula derived for mass of almost Kähler manifolds, extending previous results.
problem Calculating mass in almost Kähler geometry.
method SpinC adaptation of Witten's proof, extending previous complex-geometric methods. result Explicit formula for ADM mass in terms of Hermitian scalar curvature and topological data.
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
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.
Study on Einstein manifolds linking stability and rigidity.
problem Einstein manifold rigidity and stability.
method Review of linear and dynamical stability, scalar curvature rigidity.
result Relation between stability and rigidity of Einstein manifolds.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
Study the positive mass theorem for certain asymptotic manifolds.
problem Positive mass theorem for specific types of asymptotic manifolds.
method Compactification and analysis of Riemannian manifolds.
result Established conditions for the positive mass theorem.
Explains how to prove positive mass theorem with boundary in dimensions less than 8.
problem Proving the positive mass theorem with boundary conditions.
method Uses established results to prove various versions of the theorem.
result Various versions of the positive mass theorem are proven for initial data sets with boundary in dimensions less than 8.
In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes fro…
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.
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…
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.
In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime [16,17], the anti-de Sitter spacetime [4], or the Schwarzschild spacetime [3]. In each case, the reference spacetime admits a conformal Killing-Yano 2-form which facilitates the application of the Minkowski formula in [15] to …
In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannia…
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
Solves Jang equation for hyperboloidal data, proving positive mass theorem.
problem Proving the positive mass theorem in asymptotically hyperbolic 3D spacetimes.
method Solves Jang equation with hyperboloidal initial data, applies to positive mass theorem.
result Non-spinor proof of positive mass theorem in 3D asymptotically hyperbolic spacetimes.
Researchers prove positive mass theorem for manifolds with arbitrary ends.
problem Proving the positive mass theorem for manifolds with non-compact ends.
method Developed techniques to handle non-compact singular sets and used Wloc1,p metrics. result Established positive mass theorem for C0 arbitrary ends with Wloc1,p metrics. Stability of positive mass theorem proven under Ricci curvature bounds.
problem Stability of positive mass theorem under Ricci curvature lower bounds.
method Harmonic level set approach combined with techniques from almost splitting theorem.
result Proves Gromov-Hausdorff stability of positive mass theorem.
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.
The paper proves density and positive mass theorems for incomplete manifolds.
problem Proving density and positive mass theorems for manifolds with incomplete ends.
method Using harmonic asymptotics and quantitative positive mass theorem improvements.
result Improved quantitative positive mass theorem in dimensions 3 to 7.
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.
We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space Wloc2,n/2(M). We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.