Schwarzschild Manifold Research Articles

A Rigidity Theorem for Surfaces in Schwarzschild Manifold

Abstract In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

On a Minkowski-like inequality for asymptotically flat static manifolds

The Minkowski inequality is a classical inequality in differential geometry giving a bound from below on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R n \mathbb {R}^n ; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdS-Schwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.

Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

In this paper, we introduce a nonlinear ODE method to construct constant mean curvature (CMC) surfaces in Riemannian manifolds with symmetry. As an application, we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like [Formula: see text]. The existence of unstable CMC spheres tells us that the stability condition in Qing–Tian’s work [On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc. 20(4) (2007) 1091–1110] cannot be removed generally.

On inverse mean curvature flow in Schwarzschild space and Kottler space

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface $\Sigma_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].

Asymptotically flat extensions of CMC Bartnik data

Let g be a metric on the 2-sphere with positive Gaussian curvature and H be a positive constant. Under suitable conditions on (g, H), we construct smooth, asymptotically flat 3-manifolds M with non-negative scalar curvature, with outer-minimizing boundary isometric to and having mean curvature H, such that near infinity M is isometric to a spatial Schwarzschild manifold whose mass m can be made arbitrarily close to a constant multiple of the Hawking mass of . Moreover, this constant multiplicative factor depends only on (g, H) and tends to 1 as H tends to 0. The result provides a new upper bound of the Bartnik mass associated with such boundary data.

Loop gravity and Schwarzschild spacetime

A discretization for the Schwarzschild spacetime manifold is introduced and investigated. It is shown that the discreteness of the area of space shown in loop gravity leads to a tetrahedral structure characterizing the Schwarzschild manifold.

Isometric embeddings of 2-spheres into Schwarzschild manifolds

Let g be a Riemannian metric on the 2-sphere S2. Results on isometric embeddings of (S2, g) into a fixed model manifold often have implications in quasi-local mass related problems in general relativity. In this paper, motivated by the definitions of the Brown–York and the Wang–Yau mass, we consider isometric embeddings of (S2, g) into conformally flat spaces. We prove that if g is close to the standard metric on S2, then (S2, g) admits an isometric embedding into any spatial Schwarzschild manifold with small mass. We also give a sufficient condition that ensures isometric embeddings of perturbations of a Euclidean convex surface into \({\mathbb{R}^3}\) equipped with a conformally flat metric.

A note on Weingarten hypersurfaces in the warped product manifold

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

Isoperimetric and Weingarten surfaces in the Schwarzschild manifold

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric surfaces for very large enclosed volume. This complements work in H. Bray’s thesis, where isoperimetric surfaces homologous to the horizon are studied.

A property of the Brown–York mass in Schwarzschild manifolds

We will extend partially our previous results about the limit of the Brown–York mass of a family of convex revolution surfaces in the Schwarzschild manifold such that these surfaces may have unbounded ratios of their radii.

The Brown–York Mass of Revolution Surfaces in Asymptotically Schwarzschild Manifolds

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.

A note on the Yamabe constant of an outermost minimal hypersurface

Using an elementary argument we find an upper bound on the Yamabe constant of the outermost minimal hypersurface of an asymptotically flat manifold with nonnegative scalar curvature that satisfies the Riemannian Penrose Inequality. Provided the manifold satisfies the Riemannian Penrose Inequality with rigidity, we show that equality holds in the inequality if and only if the manifold is the Riemannian Schwarzschild manifold.

Gravitational waves as deformations of embedded Einstein spaces

We show how approximate radiative solutions of Einstein's equations can be constructed using small deformations of Einstein spacetimes embedded into a pseudo-Euclidean flat space of higher dimension. Infinitesimal deformations are seen then as vector fields in EN. All geometrical quantities can be then expressed in terms of embedding functions zA and their deformations . Then we require the deformations to keep Einstein equations satisfied up to a given order in ε. The system obtained is then analyzed in particular cases of the Minkowski and Schwarzschild manifolds taken as a starting point, and solutions of deformations of Einstein's equations displaying radiative behavior are found up to the third order of expansion in small parameter ε.

Thin shell dynamics and equations of state

A relation between stress-energy and motion is derived for accelerated Israel layers. The relation, for layers between two Schwarzschild manifolds, generalizes the equation of state for geodesic collapse. A set of linked layers is discussed.

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

We consider the wave equation (-\dt^2+\dr^2 -V -V_L(-\Delta_{S^2})) u = fF'(|u| ^2) u with (t,\rho,\theta,\phi) in R x R x S^2. The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in L^2_{loc} for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V_L, and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.

Naked shell singularities on the brane

By utilizing nonstandard slicings of 5-dimensional Schwarzschild and Schwarzschild-AdS manifolds based on isotropic coordinates, we generate static and spherically-symmetric braneworld spacetimes containing shell-like naked null singularities. For planar slicings, we find that the brane-matter sourcing the solution is a perfect fluid with an exotic equation of state and a pressure singularity where the brane crosses the bulk horizon. From a relativistic point of view, such a singularity is required to maintain matter infinitesimally above the surface of a black hole. From the point of view of the AdS/CFT conjecture, the singular horizon can be seen as one possible quantum correction to a classical black hole geometry. Various generalizations of planar slicings are also considered for a Ricci-flat bulk, and we find that singular horizons and exotic matter distributions are common features.

Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates

The semilinear wave equation on the (outer) Schwarzschild manifold is studied. We prove local decay estimates for general (non-radial) data, deriving a-priori Morawetz type estimates.

Geometric interpretation of Schwarzschild instantons

We address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L 2 harmonic 2-forms on the space. Gibbons found a non-topological L 2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction in the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L 2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2 n 2 found by Charap and Duff in 1977. Using results of Dodziuk and Hitchin we also calculate the full L 2 harmonic space for the Euclidean Schwarzschild manifold.

Canonical quantization inside the Schwarzschild black hole

We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to CPT invariance to construct an explicitly positive-definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical metric. We also propose an expression for the propagator in the interior and express it as a mode sum. The field theory thus obtained is meaningful for small curvatures far from the classical singularity.

Kaluza - Klein electrically charged black branes in M-theory

We present a class of Kaluza - Klein electrically charged black p-brane solutions of 10-dimensional, type IIA superstring theory. Uplifting to 11 dimensions these solutions are studied in the context of M-theory. They can be interpreted either as a p + 1 extended object trapped around the 11th dimension along which momentum is flowing or as a boost of the following backgrounds: the Schwarzschild black (p + 1)-brane or the product of the (10-p)-dimensional Euclidean Schwarzschild manifold with the (p + 1)-dimensional Minkowski spacetime.