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.
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 paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.
problem Investigating the stability of mean curvature flows in specific spacetime geometries.
method Combining center manifold analysis with global existence results for flows near isoperimetric hypersurfaces.
result Global existence and convergence to constant mean curvature (CMC) hypersurfaces for flows in asymptotic Schwarzschild space.
We study the problem of existence of isoperimetric regions for large volumes, in C0-locally asymptotically Euclidean Riemannian manifolds with a finite number of C0-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in …
In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.
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. 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.
The paper proves the stability of a flow in Schwarzschild space.
problem Stability of area preserving mean curvature flow in asymptotic Schwarzschild space.
method Demonstrates existence and exponential convergence of the flow for all time.
result The flow converges to a round sphere or a constant mean curvature surface.
Study of area preserving Willmore flow on Schwarzschild-like manifolds.
problem Stability of Willmore spheres in Schwarzschild-like ends.
method Area preserving Willmore flow analysis in C3−close Schwarzschild ends. result Leaves of the Willmore foliation are strict local area preserving maximizers of Hawking mass.
Study on area-constrained Willmore spheres in asymptotic Schwarzschild manifolds.
problem Existence of area-constrained Willmore spheres with non-negative Hawking mass and inner radius.
method Analysis of scalar curvature and asymptotic properties of 3-manifolds.
result No large area-constrained Willmore spheres exist under certain conditions.
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 paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.
problem Understanding Willmore surfaces in asymptotically Schwarzschild 3-manifolds.
method Application of Lyapunov-Schmidt reduction method.
result End of the manifold is foliated by area-constrained Willmore spheres.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
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.
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 finds existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
problem Existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
method Extends Lyapunov-Schmidt analysis to 'far-off-center' regime and general Schwarzschild asymptotics.
result Sharp existence and non-existence results for large stable CMC spheres.
Researchers found unique large stable spheres in specific 3D space.
problem Characterizing large stable spheres in specific types of 3D space.
method Unconditional characterization of spheres using Riemannian geometry.
result Global uniqueness of large stable spheres in asymptotically flat Riemannian three-manifolds.
We derive a weighted L2-estimate of the Witten spinor in a complete Riemannian spin manifold (Mn,g) of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of M enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactifi…
The paper establishes inequalities for p-capacitary functions in flat half-spaces.
problem Understanding p-capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
Proves equality in Minkowski inequality for static, flat manifolds.
problem Proving equality in Minkowski inequality for static, flat manifolds.
method Analyzes quasi-spherical metrics and static manifolds.
result Equality in Minkowski inequality achieved only by Schwarzschild space slices.
We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically …
Study linear perturbations in Schwarzschild black hole spacetime.
problem Linear perturbations of Schwarzschild black hole spacetime.
method Investigate linearised perturbation of constant mass aspect function foliation at null infinity.
result Linearised perturbations of Bondi energy and mass vanish, and all linear momentum can be achieved.
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.
We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.
This work characterizes asymptotically hyperbolic 3-manifolds via CMC-foliations.
problem Characterizing asymptotically hyperbolic 3-manifolds.
method Existence of a suitable CMC-foliation with geometric curvature estimates.
result Any 3-manifold is asymptotically hyperbolic if it has a CMC-cover with controlled instability.
The paper characterizes photon surfaces in static spacetimes and proves their uniqueness.
problem Characterizing and proving uniqueness of photon surfaces in static spacetimes.
method Local characterization and proof of uniqueness using specific spacetime properties.
result Static, vacuum, asymptotically isotropic spacetimes with equipotential photon surfaces are isometric to Schwarzschild spacetime.
Proves a Minkowski-like inequality for static flat manifolds.
problem Bounding total mean curvature of convex surfaces.
method Adapting Wei's analysis to static asymptotically flat manifolds.
result Proves a Minkowski-like inequality for general static asymptotically flat manifolds.
Constructs surfaces with constant mean curvature in Schwarzschild spacetime near null infinity.
problem Creating surfaces with constant mean curvature in the Schwarzschild spacetime near null infinity.
method Constructs spacelike surfaces as graphs of functions u(y, r) with specified asymptotic behavior.
result The constructed surfaces intersect future null infinity with a cut given by a function f.
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.
Study peels tensor equations on Schwarzschild spacetime.
problem Analyzing the asymptotic behavior of tensorial wave equations on Schwarzschild spacetime.
method Combining conformal compactification and vector field techniques to estimate tensorial field energies.
result Obtains optimal initial data for peeling at all orders.
Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.
Paper proves rigidity of static manifolds and applies to metric extensions.
problem Detecting rotational symmetry in static systems.
method Conformal techniques and Minkowski-type inequalities.
result Proves global uniqueness of static metric extensions.
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…
Embeds spherically symmetric space-times into 5D flat space, matching Hawking temperature.
problem Finding conditions for embedding spherically symmetric space-times into flat 5D space.
method Explicitly constructs embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions.
result Hawking temperature matches Unruh temperature for Schwarzschild metric.
The paper constructs unstable and outlying CMC spheres in specific manifolds.
problem Stability of CMC spheres in certain manifolds.
method Non-linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry.
result Existence of unstable CMC spheres, indicating the stability condition cannot be removed.
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.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
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.
The paper proves the Null Penrose Inequality for Schwarzschild spacetimes.
problem Proving the Null Penrose Inequality for Schwarzschild spacetimes.
method Introducing quasi-round MOTSs and showing their stability and existence for specific horizons.
result Identifies sufficient conditions for the Null Penrose Inequality in Schwarzschild spacetimes.
Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.
problem Analyzing spacelike CMC surfaces near future null-infinity in Schwarzschild spacetime.
method Proves asymptotic hyperbolicity, derives boundary data expressions, and shows compatibility conditions.
result Compatibility conditions and asymptotic behaviors of spacelike CMC surfaces near future null-infinity.
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.
We will discuss existence of center of mass on asymptotically Schwarzschild manifold defined by Huisken-Yau and Corvino-Schoen. Conditions of existence and examples on non existence are given.
Constructs 3-manifolds with flat ends and positive mean curvature.
problem Finding asymptotically flat 3-manifolds with specific boundary conditions.
method Constructs smooth, asymptotically flat 3-manifolds with non-negative scalar curvature.
result Provides a new upper bound for the Bartnik mass.
We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.
We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological consta…
Penrose inequality proven for Schwarzschild black hole perturbations.
problem Proving Penrose inequality for black hole perturbations.
method Perturbative approach, deforming null hypersurface to ensure monotonically increasing Hawking mass and asymptotically round leaves.
result Lower bound on ADM energy for Schwarzschild black hole perturbations.
Proves Penrose inequality for cohomogeneity one initial data sets.
problem Proving Penrose inequality for specific initial data sets.
method Analyzing asymptotically flat and hyperbolic initial data sets under cohomogeneity one actions.
result Total mass is bounded below by a function of outermost apparent horizon area, with equality for Schwarzschild(-AdS) embeddings.