我们考虑在具有正ADM能量的渐近欧几里得初始数据集中，曲面的体积保持曲率演化。速度由平均曲率的非线性函数给出，该函数推广了Cederbaum-Sakovich（Calc. Var. PDE, 2021）最近考虑的时空平均曲率。遵循Huisken-Yau（Invent. Math., 1996）的经典方法，我们证明了从适当圆形的初始曲面出发的流对所有时间存在，并收敛到一个常数（时空）曲率极限。这提供了Cederbaum-Sakovich的CSTMC叶状结构的另一种构造，并在广义相对论中孤立系统的质心定义中有应用。
英文摘要：
We consider a volume preserving curvature evolution of surfaces in an asymptotically Euclidean initial data set with positive ADM-energy. The speed is given by a nonlinear function of the mean curvature which generalizes the spacetime mean curvature recently considered by Cederbaum-Sakovich (Calc. Var. PDE, 2021). Following a classical approach by Huisken-Yau (Invent. Math., 1996), we show that the flow starting from suitably round initial surfaces exists for all times and converges to a constant (spacetime) curvature limit. This provides an alternative construction of the CSTMC foliation by Cederbaum-Sakovich and has applications in the definition of center of mass of an isolated system in General Relativity.
We solve the spacelike, spherically symmetric, constant mean curvature hypersurfaces in the maximally extended Reissner-Nordstrom spacetime with the charge smaller than the mass. Based on these results, we construct constant mean curvature foliations with fixed or varied mean curvature in each slice in this spacetime.
We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotica…
We consider open globally hyperbolic spacetimes $N$ of dimension $n+1$, $n\ge 3$, which are spatially asymptotic to a Robertson-Walker spacetime or an open Friedmann universe with spatial curvature $\tildeκ= 0,-1$ and prove, under reasonable assumptions, that there exists a unique foliation by hypersurfaces of constant mean curvature and that the mean curvature function $τ$ is a smooth time function if $N$ is smooth. Moreover, among the Friedmann universes which satisfy the necessary conditions are those that reflect the present assumptions of the development of the universe.
We identify certain general geometric conditions on a foliation of a spacetime (M,g) by timelike curves that will impede the existence of null geodesic lines, especially if (M,g) possesses a compact Cauchy hypersurface. The absence of such lines, in turn, yields well-known restrictions on the geometry of cosmological spacetimes, in the context of Bartnik's splitting conjecture. Since the (non)existence of null lines is actually a conformally invariant property, such conditions only need to apply for some suitable conformal rescaling of g.
We study the linearised perturbation of the constant mass aspect function foliation in a Schwarzschild black hole spacetime. In particular, we investigate the linearised perturbation of the asymptotic geometry of the foliation at null infinity. We show that there is a 4-dimensional linear space for the linearised perturbation of the initial leaf inside the event horizon, corresponding to which the linearised perturbation of the asymptotic geometry of the foliation at null infinity is preserved to be round. For such a linearised perturbation of the initial leaf in this 4-dimensional linear spac…
A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of such an equation are obtained. In particular, the case of entire spacelike graphs in de Sitter spacetime is faced and completely solved by means of a new application of a known integral formula.
The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of …
This paper addresses pure gauge questions in the study of (asymptotically) de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results which are relevant for the foliations used in the stability problem of the expanding region of Schwarzschild de Sitter spacetimes [Schlue (2016)]: (i) Small perturbations of round spheres on the cosmological horizons lead to spheres that pinch off at infinity. (ii) Globally well behaved…
In this paper, we introduce the notion of a marginal tube, which is a hypersurface foliated by marginal surfaces. It generalises the notion of a marginally trapped tube and several notions of black hole horizons, for example trapping horizons, isolated horizons, dynamical horizons, etc. We prove that if every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null. There is no assumption on the topology of the marginal tube. To prove it, we study the geometry of spacelike surfaces in a 4-dimensional spacetime with the help of double null coordinate systems. T…
We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constant scalar curvature. Furthermore, this time function is a smooth submersion.
In this paper, we introduce an entropy functional on Riemannian foliation, inspired by the work of , which is monotonically along the transverse Ricci flow. We relate their gradient flow, via diffeomorphism preserving the foliated structure of the manifold with Riemannian foliation, to the transverse Ricci flow. Moreover, inspired by the work of Fuquan Fang and Yuhao Zhang, we give a necessary condition for codimension 4 Riemannian foliation admitting the transverse Einstein metric.
We consider expanding vacuum spacetimes with a CMC foliation by compact spacelike hypersurfaces. Under scale invariant a priori geometric bounds (type-III), we show that there are arbitrarily large future time intervals that are modelled by a flat spacetime or a Kasner spacetime. We give related results for a class of expanding vacuum spacetimes that do not satisfy the a priori bounds (type-II).
