Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
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48 results for “mass rigidity”
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.
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.
Rigidity results for initial data sets related to the positive mass theorem.
problem Rigidity of initial data sets in general relativity.
method Establishing conditions for weak outermost marginally outer trapped surfaces and rigidity results for Riemannian manifolds.
result Marginally outer trapped surfaces are weakly outermost under certain conditions.
Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifol…
It is well-know that Hawking mass is nonnegative for a stable constant mean curvature () sphere in three manifold of nonnegative scalar curvature. R. Bartnik proposed the rigidity problem of Hawking mass of stable spheres. In this paper, we show partial rigidity results of Hawking mass for stable spher…
Schoen-Yau's zero mass theorem stability remains an open question.
problem Geometric stability of Schoen-Yau's zero mass theorem.
method Review of geometric stability, examples, and convergence notions.
result Open question on geometric stability of Schoen-Yau's zero mass theorem.
We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds or under special asymptotics.
New formulas for hyperbolic mass using horospheres.
problem Mass calculation of asymptotically hyperbolic manifolds.
method Geometric formulas derived using coordinate horospheres.
result Improved rigidity results of hyperbolic space.
In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects t…
Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.
problem Rigidity of 3-manifolds with boundary under specific geometric conditions.
method Area estimates for free boundary strictly stable two-disks, modified Hawking mass analysis.
result 3-manifolds with boundary are locally isometric to half anti-de Sitter-Schwarzschild manifold.
Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
method Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
result Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
Rigidity results for Hawking mass in curved spaces with bounds on Bartnik capacity.
problem Rigidity of surfaces in curved spaces with mass bounds.
method Analyzing Hawking mass and applying rigidity results to specific geometric settings.
result Explicit lower bounds on Hawking and Bartnik masses in non-flat spaces.
Paper proves rigidity of CMC surfaces in curved 3-manifolds.
problem Rigidity of CMC surfaces in positive curved 3-manifolds.
method Assumptions of surface being approximately round or invariant under even symmetry, and use of Hawking mass.
result Rigidity results for stable CMC surfaces with zero Hawking mass.
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…
Formula derived for mass of almost Kähler manifolds, extending previous results.
problem Calculating mass in almost Kähler geometry.
method Spin 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.
Einstein manifolds are rigid under certain metric deformations.
problem Characterizing Einstein manifolds that resist volume-preserving metric deformations.
method Various characterizations and constructions of mass-decreasing perturbations.
result Constructs mass-decreasing perturbations of specific metrics.
Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.
problem Positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries.
method Spinorial proof and optimal lower bound for eigenvalues.
result Optimal lower bound for first non-null eigenvalue of Dirac operator.
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.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Study shows rigidity of polyhedrons in hyperbolic spaces.
problem Rigidity of polyhedrons in hyperbolic spaces.
method Extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds.
result Localization of the positive mass theorem for asymptotically hyperbolic manifolds.
New rigidity result for metrics with positive scalar curvature and specific decay.
problem Understanding metrics with positive scalar curvature and decay.
method Analyzing metrics with non-negative scalar curvature and decay properties.
result Metrics with specific decay properties must be flat.
In this paper we show how a natural coupling of the Dirac equation with the generalized Jang equation, leads to a proof of the rigidity statement in the positive mass theorem with charge, without the maximal slicing condition, provided a solution to the coupled system exists.
Study rigidity of minimal disks in specific 3-manifolds.
problem Rigidity of free boundary minimal disks in mean convex three-manifolds.
method Assuming strict stability, prove isometric neighborhoods using modified Hawking mass.
result Prove rigidity of minimal disks in specific 3-manifolds.
In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.
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.
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.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
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.
In this paper, we prove that the even solution of the mean field equation on must be axially symmetric when . In particular, zero is the only even solution for . This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.
Green functions for GJMS operators on spheres derived, linking geometry and rigidity.
problem Deriving Green functions for GJMS operators on spheres.
method Explicit representation formulae derived using Gegenbauer polynomials.
result Spheres uniquely characterized by their Green functions, with strong rigidity theorems for .
The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.
problem Proving a positive mass theorem for non-compact static domains in hyperbolic space.
method Formulating and proving a positive mass theorem under natural dominant energy conditions, using elliptic boundary conditions on spinors.
result Retrieve a sharper version of a recent result by Souam about the rigidity of non-compact static domains.
In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.
Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean spacemath.DG
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…
Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface , we observe that …
In this short note, we prove positivity of Brown-York mass under quasi-positive boundary data which generalize some previous results by the authors. The corresponding rigidity result is obtained.
In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only const…
We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a point or orthogonal to an hypersurface, one of these metrics being a warped product if the dimension is greater than or equal to 3. This formula has non-trivial geometric consequences such as a positive mass type t…
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.
Spectral Positive Mass Theorem for Asymptotically Hyperbolic 3-manifolds with Toroidal Infinitymath.DG
Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
problem Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
method Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
result Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
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.
The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.
problem Determining if a spacetime is conformally AdS based on null geodesic travel times.
method Analyzing all null geodesics from a point to its antipodal point, considering various spacetime conditions.
result The spacetime is conformally AdS if and only if all null geodesics from a point refocus at its antipodal point.
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.
Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asympto…
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.
Proves rigidity of sphere metrics with subsets removed.
problem Scalar curvature rigidity of spheres with subsets removed.
method Techniques involving wrapping property and metrics.
result Proves scalar rigidity for metrics on .