Constant Mean Curvature Foliations Research Articles

Constant mean curvature foliation of the Taub–NUT metric

Constant mean curvature foliation of the Taub–NUT metric

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

A Note on Constant Mean Curvature Foliations of Noncompact Riemannian Manifolds

We aimed to study constant mean curvature foliations of noncompact Riemannian manifolds, satisfying some geometric constraints. As a byproduct, we answer a question by M. P. do Carmo (see Introduction) about the leaves of such foliations.

Constant mean curvature foliation of globally hyperbolic (2 $$+$$ + 1)-spacetimes with particles

Let M be a globally hyperbolic maximal compact 3-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that M admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularities of angles less than $$\pi $$ along time-like geodesics).

Constant mean curvature foliation of domains of dependence in 𝐴𝑑𝑆₃

We prove that, given an acausal curve Γ \Gamma in the boundary at infinity of A d S 3 AdS_{3} which is the graph of a quasi-symmetric homeomorphism ϕ \phi , there exists a unique foliation of its domain of dependence D ( Γ ) D(\Gamma ) by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of ϕ \phi .

Hamiltonian analysis of the cuscuton

The cuscuton was introduced in the context of cosmology as a field with infinite speed of propagation. It has been claimed to resemble Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in a certain limit, and it is a good candidate for an ether theory in which a time-dependent cosmological constant appears naturally. The analysis of its properties is usually performed in the Lagrangian framework, which makes issues like the counting of its dynamical degrees of freedom less clear-cut. Here we perform a Hamiltonian analysis of the theory. We show that the homogeneous limit with local degrees of freedom has singular behavior in the Hamiltonian framework. In other frames, it has an extra scalar degree of freedom. The homogeneous field has regular behavior only if defined a priori as a spatially constant field in a constant mean curvature foliation and contributing with a single global degree of freedom. Lastly, we find conditions on the cuscuton potential for the resulting lapse function to be nonzero throughout evolution.

Existence of a constant mean curvature foliation in the extended Schwarzschild spacetime

We construct a $T$-axisymmetric, spacelike, spherically symmetric, constant mean curvature hypersurfaces foliation in the Kruskal extension with properties that the mean curvature varies in each slice and ranges from minus infinity to plus infinity. This family of hypersurfaces extends the CMC foliation discussions posted by Malec and \'{O} Murchadha in 2009.

CMC Foliations of Closed Manifolds

We prove that every closed, smooth \(n\)-manifold \(X\) admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of \(X\) can be chosen so that when \(n\ge 2\), the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of \(X\) that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.

Explicit Riemannian Manifolds with Unexpectedly Behaving Center of Mass

The (relativistic) center of mass of an asymptotically flat Rie- mannian manifold is often defined by certain surface integral expressions eval- uated along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and Misner (ADM). There are also what we call abstract definitions of the center of mass in terms of a foliation near infinity itself, going back to the constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give rise to surface integral expressions when equipped with suitable systems of coordinates. We discuss subtle asymptotic convergence issues regarding the ADM- and the coordinate expressions related to the CMC-center of mass. In particular, we give explicit examples demonstrating that both can diverge - in a setting where Einstein's equation is satisfied. We also give explicit exam- ples of the same asymptotic order of decay with prescribed mass and center of mass. We illustrate both phenomena by providing analogous examples in Newtonian gravity. Our examples conflict with some results in the literature.

Dynamical emergence of universal horizons during the formation of black holes

Motivations for the existence of a fundamental preferred frame range from pure phenomenology to attempts to solve the non-renormalizability of quantum gravity, the problem of time (and scale), and the cosmological constant problem(s). In many explicit constructions, such as Einstein-Aether or Gravitational Aether theories, K-essence, Cuscuton theory, Shape Dynamics, or (non-projectable) Horava-Lifshitz gravity, the low energy theory contains a fluid (which defines a preferred frame) with superluminal or incompressible excitations. We study here the formation of black holes in the presence of such a fluid. In particular, we focus on the incompressible limit of the fluid (or Constant Mean Curvature foliation) in the space-time of a spherically collapsing shell within an asymptotically cosmological space-time. In this case, ignoring the fluid back reaction, we can analytically show that an observer inside 3/4 of the Schwarzschild radius cannot send a signal outside, after a stage in collapse, even using signals that propagate infinitely fast in the preferred frame. This confirms the dynamical emergence of universal horizons that have been previously found in static solutions. We argue that this universal horizon should be considered as the future boundary of the classical space-time.

Axisymmetric constant mean curvature slices in the Kerr spacetime

Recently, there have been efforts to solve Einstein’s equation in the context of a conformal compactification of spacetime. Of particular importance in this regard are the so-called constant mean curvature (CMC) foliations, characterized by spatial hyperboloidal hypersurfaces with a constant extrinsic mean curvature K. However, although of interest for general spacetimes, CMC slices are known explicitly only for the spherically symmetric Schwarzschild metric. This work is devoted to numerically determining axisymmetric CMC slices within the Kerr solution. We construct such slices outside the black hole horizon through an appropriate coordinate transformation in which an unknown auxiliary function A is involved. The condition K = const throughout the slice leads to a nonlinear partial differential equation for the function A, which is solved with a pseudo-spectral method. The results exhibit exponential convergence, as is to be expected in a pseudo-spectral scheme for analytic solutions. As a by-product, we identify CMC slices of the Schwarzschild solution which are not spherically symmetric.

Decomposition of gravitational perturbations on generic backgrounds with constant mean curvature foliations

A decomposition of linear gravitational perturbations in 1 + d form is presented which differs in several respects from earlier ones. (i) The boundary value problem with data on initial and final surface is treated for generic background spacetimes with constant mean curvature (CMC) foliations thereby avoiding issues with linearization instability. (ii) Gauge invariant counterparts of the linearized lapse, shift, and spatial metric are constructed via gauge fixing such that the quadratic action has the same functional dependence on them as on the original fields. (iii) The linearized CMC condition is shown to provide a gauge condition for the temporal gauge transformations separately such that the physical degrees of freedom can be chosen to reside entirely in the linearized spatial metric. (iv) In such gauges the linearized Hamiltonian and diffeomorphism constraints decouple and can be solved by inversion of elliptic differential operators also within the action. (v) An isometry to Moncrief’s three term orthogonal but doubly coupled phase space decomposition can be set up.

Constant curvature foliations in asymptotically hyperbolic spaces

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\partial M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $\sigma_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.

Constant mean curvature slicings of Kantowski-Sachs spacetimes

We investigate existence, uniqueness, and the asymptotic properties of constant mean curvature (CMC) slicings in vacuum Kantowski-Sachs spacetimes with positive cosmological constant. Since these spacetimes violate the strong energy condition, most of the general theorems on CMC slicings do not apply. Although there are in fact Kantowski-Sachs spacetimes with a unique CMC foliation or CMC time function, we prove that there also exist Kantowski-Sachs spacetimes with an arbitrary number of (families of) CMC slicings. The properties of these slicings are analyzed in some detail.

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge–Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken–Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied.

Evolution of a massless test scalar field on boson star space-times

We numerically solve the massless test scalar field equation on the space-time background of boson stars and black holes. In order to do so, we use a numerical domain that contains future null infinity. We achieve this construction using a scri-fixing conformal compactification technique based on hyperboloidal constant mean curvature foliations of the space-time and solve the conformally invariant wave equation. We present two results: the scalar field shows oscillations of the quasinormal mode type found for black holes only for boson star configurations that are compact; and no signs of tail decay are found in the parameter space we explored. Even though our results do not correspond to the master equation of perturbations of boson star solutions, they indicate that the parameter space of boson stars as black hole mimickers is restricted to compact configurations.

Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds II

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.

Existence and Uniqueness of Constant Mean Curvature Foliation of Asymptotically Hyperbolic 3-Manifolds

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 constant.

General spherically symmetric constant mean curvature foliations of the Schwarzschild solution

We consider a family of spherical three-dimensional spacelike slices embedded in the Schwarzschild solution. The mean curvature is constant on each slice but can change from slice to slice. We give a simple expression for an everywhere positive lapse and thus we show how to construct foliations. There is a barrier preventing the mean curvature from becoming large, and we show how to avoid this so as to construct a foliation where the mean curvature runs all the way from zero to infinity. No foliation exists where the mean curvature goes from minus to plus infinity. There are slicings, however, where each slice passes through the bifurcation sphere $R=2M$ and the lapse only vanishes at this one point, and is positive everywhere else, while the mean curvature does run from minus to plus infinity. Symmetric foliations of the extended Schwarzschild spacetime degenerate at a critical point, where we show that the lapse function exponentially approaches zero.

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.