Study on linear stability of Schwarzschild spacetime using harmonic gauge.
problem Linear stability of Schwarzschild spacetime in harmonic gauge.
method Analysis of odd solution of linearized Einstein equation using Regge-Wheeler quantities.
result Solution decays at rate τ^(-1+δ) to a linearized Kerr solution.
Schwarzschild black hole stability proven using a new gauge.
problem Linear stability of Schwarzschild black holes to gravitational perturbations.
method Proved stability by imposing a generalised wave gauge.
result Uniform boundedness and polynomial decay of solutions.
Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.
problem Stability of the Schwarzschild singularity in near-Schwarzschild black holes.
method Energy methods and new approach to Einstein vacuum equations in axial symmetry.
result The solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity.
Higher-dimensional Schwarzschild spacetimes violate the Penrose property.
problem Causal behavior of higher-dimensional Schwarzschild spacetimes.
method Analyzing causal properties in (2+1), (3+1), and (d+1) dimensions. result The Penrose property does not hold for (d+1) dimensional Schwarzschild if d>3. In this paper, we prove the existence of an ancient solution to the Ricci flow whose limit at t=−∞ is the Euclidean Schwarzschild metric.
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
Inverse curvature flows in AdS-Schwarzschild manifold converge to a sphere.
problem Understanding the behavior of hypersurfaces in AdS-Schwarzschild geometry.
method Non-parametric inverse curvature flows with 1-homogeneous curvature function.
result Principle curvatures converge to 1 exponentially fast.
Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.
Proves stability of Schwarzschild black holes without symmetry assumptions.
problem Stability of Schwarzschild black holes under general conditions.
method Teleologically normalised double null gauges, analysis of linear stability, and control of non-linearities.
result Proves non-linear asymptotic stability of Schwarzschild family as solutions to Einstein vacuum equations.
Proves a key inequality for special geometric shapes in space-time.
problem Proving a Minkowski-type inequality for specific geometric shapes.
method Using weak solution of Inverse mean curvature flow.
result Sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.
Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.
problem Existence of asymptotically hyperbolic solutions to Einstein constraints with marginally outer trapped boundaries.
method Constant mean curvature conformal method.
result Verification of Penrose inequality for certain Schwarzschild-AdS black hole perturbations.
Study shows stability of Schwarzschild spacetime under specific perturbations.
problem Linear stability of Schwarzschild spacetime under axial perturbations.
method Complex line bundle interpretation and connection-level object analysis.
result Suitably regular initial data decay to a linearized Kerr metric.
Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
Maxwell equations decay to Coulomb solutions on black hole spacetimes.
problem Decay of Maxwell solutions in Schwarzschild-de Sitter spacetimes.
method Differential transformation of Maxwell tensor components, Fackerell-Ipser equation, vector field method.
result Super-polynomial decay rate of Maxwell solutions to Coulomb solutions.
In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.
Proves stability of Schwarzschild black holes, showing metric coefficients decay.
problem Linear stability of Schwarzschild black holes in vacuum Einstein equations.
method Linearized gravity, Hodge decomposition, gauge-invariant master quantities, decay estimates.
result Metric coefficients decay to linearized Kerr metric on exterior region.
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
New instantons found on Euclidean Schwarzschild manifold, challenging existing theories.
problem Challenging the understanding of instantons on the Euclidean Schwarzschild manifold.
method Exploring a correspondence between planar Abelian vortices and spherically symmetric instantons.
result Complete description of a connected component of the moduli space of unit energy SU(2) instantons.
Study linearized Schwarzschild spacetimes, proving decay of master quantities.
problem Linear stability of higher-dimensional Schwarzschild spacetimes.
method Hodge decomposition, gauge-invariant master quantities, wave equations.
result Uniform boundedness and decay estimates for master quantities in 6 or fewer dimensions.
The paper examines curvature properties of Vaidya metric, a generalization of Schwarzschild solution.
problem Investigating curvature properties of Vaidya metric.
method Analyzing Vaidya metric under different curvature conditions and comparing with Ludwig-Edgar pure radiation metric.
result Vaidya metric exhibits various curvature properties including pseudosymmetric type conditions and is Ricci simple.
Minimal surfaces in Schwarzschild manifold have Morse index one.
problem Existence of unbounded minimal surfaces in asymptotically flat 3-manifolds.
method Analysis of Riemannian Schwarzschild manifold, free boundary condition.
result Minimal surface in Schwarzschild manifold has Morse index one.
New theorem proves uniqueness of Schwarzschild-de Sitter spacetime.
problem Proving the uniqueness of Schwarzschild-de Sitter spacetime.
method Developed new tools like reverse Lojasiewicz inequality and proved smoothness of maximum-set of lapse.
result Established a new uniqueness theorem for three-dimensional Schwarzschild-de Sitter metrics.
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…
Classifies static vacuum black holes in 3+1 dimensions.
problem Uniqueness of static vacuum black holes with compact horizons.
method Conformal geometry and comparison geometry techniques.
result All static vacuum black holes are either Boost, Schwarzschild, or Myers/Korotkin-Nicolai type.
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…
Study proves non-degeneracy of certain metrics in linearized gravity.
problem Proving non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics.
method Analyzing solutions of the linearized Einstein equations around Kottler metrics.
result Linearized Einstein operator is non-degenerate for open ranges of mass parameter.
New approach confirms Kruskal-Szekeres extension for Schwarzschild spacetime.
problem Confirming the Kruskal-Szekeres extension for Schwarzschild spacetime.
method Reformulating the problem as an ODE and showing the ODE admits a solution if and only if the horizon is non-degenerate.
result Photon surfaces approaching the Killing horizon must necessarily cross it.
Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.
problem Finding constant mean curvature surfaces in specific spacetimes.
method Partial differential equations in the complex plane, generalizing holomorphy.
result Extends Hopf's theorem to new spacetime geometries.
It is shown that the Schwarzschild spacetime can be extended so that the metric becomes analytic at the singularity. The singularity continues to exist, but it is made degenerate and smooth, and the infinities are removed by an appropriate choice of coordinates. A family of analytic extensions is found, and one of thes…
Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.
problem Proving a Minkowski inequality for specific types of hypersurfaces.
method Using weakly mean convex and star-shaped hypersurfaces in warped cylinders, and applying the inverse mean curvature flow.
result Sharp inequality holds for outward minimizing hypersurfaces in Schwarzschild and hyperbolic spaces.
In this global study of solutions to the linear wave equation on Schwarzschild de Sitter spacetimes we attend to the cosmological region of spacetime which is bounded in the past by cosmological horizons and to the future by a spacelike hypersurface at infinity. We prove an energy estimate capturing the expansion of th…
We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on S1×S2 with U(1)×SO(3)-invariant seed data. The resulting CMC slicings of Schwarzschild-de Sitter and Nariai are described.
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
Proves uniqueness of black holes in Lovelock gravity, a generalization of Einstein's theory.
problem Proving uniqueness of black holes in Lovelock gravity.
method Combining regularity and rigidity arguments to prove global uniqueness.
result Global uniqueness theorem for extrinsic black holes in Lovelock gravity.
In recent work, we have proven uniform decay bounds for solutions of the wave equation □gφ=0 on a Schwarzschild exterior, in particular, the uniform pointwise estimate ∣φ∣≤Cv+−1, which holds throughout the domain of outer communications, where v is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…
New method solves 'googly problem' for Schwarzschild black holes.
problem Solving the 'googly problem' for non-chiral field configurations.
method Using twistor space and holomorphic loci to construct a non-self-dual, four-dimensional Kähler metric.
result First instance of a non-self-dual Einstein metric from holomorphic data in twistor space.
Proves rigidity for surfaces in Schwarzschild manifold.
problem Rigidity of surfaces in Schwarzschild manifold.
method Variational formula of quasi-local mass.
result Proves rigidity theorem for isometric embeddings.
Paper extends foliation results in higher dimensions for Schwarzschild spaces.
problem Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds.
method Generalization to higher dimensions, proving existence under arbitrary dimensionality.
result Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds of arbitrary dimension.
Study optimizes mass bounds for black holes and de Sitter spacetimes.
problem Optimizing mass bounds for static metrics with positive cosmological constant.
method Rigorous mathematical proofs of area bounds and rigidity statements.
result Proves optimal area bounds for horizons of black hole and cosmological types.
We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci …
Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
We consider solutions to the linear wave equation □gφ=0 on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may con…
Constructs many black hole spacetimes in de Sitter space.
problem Creating well-controlled many black hole spacetimes in de Sitter space.
method Gluing Schwarzschild-de Sitter or Kerr-de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions.
result Solves the Einstein equation directly for the metric, given scattering data at the future conformal boundary.
We study Ricci flows of some classes of physically valuable solutions in Einstein and string gravity. The anholonomic frame method is applied for generic off-diagonal metric ansatz when the field/ evolution equations are transformed into exactly integrable systems of partial differential equations. The integral varieti…
Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
problem Behavior of minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
method Analysis of free boundary minimal hypersurfaces and totally geodesic hyperplanes in Schwarzschild n-manifolds. result A free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region.