Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally foliated by constant mean curvature hypersurfaces $M_τ$, with mean curvature $τ$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define the rescaled volume of $M_τ$ by $\Ham = |τ|^n \Vol(M,g)$, where $g$ is the induced metric. Then $\Ham \geq n^n \Vol(M,g_0)$ where $g_0$ is the hyperbolic metric on $M$ with sectional curvature -1. Equality holds if and only if $(M,g)$ is isometric to $(M,g_0)$.
For $2+1$ spacetime dimensions, we derive sufficient conditions for the twisting function in a twisted product spacetime, such that there is a global foliation by spacelike CMC surfaces.
An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.
We prove that the mean curvature $τ$ of the slices given by a constant mean curvature foliation can be used as a time function, i.e. $τ$ is smooth with non-vanishing gradient.
We construct 2-surfaces of prescribed mean curvature in 3-manifolds carrying asymptotically flat initial data for an isolated gravitating sysqtem with rather general decay conditions. The surfaces in question form a regular foliation of the asymptotic region of such a manifold. We recover physically relevant data, especially the ADM-momentum, from the geometry of the foliation. For a given set of data (M,g,K), with a three dimensional manifold M, its Riemannian metric g, and the second fundamental form K in the surrounding four dimensional Lorentz space time manifold, the equation we solve is …
Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivity of the isoperimetric mass established by G. Huisken we prove an existence result for large isoperimetric regions in general asymptotically flat three manifolds with non-negative scalar curvature.
A necessary and sufficient condition for the leaves of a {\em non-degenerate} foliation of a pseudo-Riemannian manifold to be conformally flat is developed. The condition mimics the classical condition of the vanishing of the Weyl or Cotton tensor establishing the conformal flatness of a pseudo-Riemannian manifold in the sense that it is also formulated in terms of the vanishing of certain tensors. These tensors play the role of the Weyl or the Cotton tensors and they are defined in terms of the the curvature of a linear torsion-free connection (the {\em bi-conformal connection}).
We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.
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 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 Ó Murchadha in 2009.
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.
In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $Ω$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference background geodesic foliation approaching large spheres and a positive function, constant along the null generators on $Ω$, which describes the relation between the two foliations at infinity. The integrand in the limit expression has interesting covariance and invariance properties wi…
We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4- and 5-manifolds with positive or nonnegative sectional curvature.
Let $M$ be a Riemannian manifold of dimension $n+1$ with smooth boundary and $p\in \partial M$. We prove that there exists a smooth foliation around $p$ whose leaves are submanifolds of dimension $n$, constant mean curvature and its arrive perpendicular to the boundary of M, provided that $p$ is a nondegenerate critical point of the mean curvature function of $\partial M$.
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres…
We study properties of the mean curvature one-form and its holomorphic and antiholomorphic cousins on a transverse Kähler foliation. If the mean curvature of the foliation is automorphic, then there are some restrictions on basic cohomology similar to that on Kähler manifolds, such as the requirement that the odd basic Betti numbers must be even. However, the full Hodge diamond structure does not apply to basic Dolbeault cohomology unless the foliation is taut.
This note provides an alternative proof of a result of Labourie. We show that the two complements of the convex core of a three dimensional quasi-fuchsian hyperbolic manifold may be foliated by embedded hypersurfaces of constant Gaussian curvature.
In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal…
We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003.
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 $\del M$ 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 $σ_k$-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.…
We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haeflig…
Let $Γ$ be a nondegenerate geodesic in a compact Riemannian manifold $M$. We prove the existence of a partial foliation of a neighbourhood of $Γ$ by CMC surfaces which are small perturbations of the geodesic tubes about $Γ$. There are gaps in this foliation, which correspond to a bifurcation phenomenon. Conversely, we also prove, under certain restrictions, that the existence of a partial CMC foliation of this type about a submanifold $Γ$ of any dimension implies that $Γ$ is minimal.
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic s…
Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation $M_τ$ in a 2+1 dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measured foliation corresponding to the translational part of the holonomy of $V$. We prove that this is the case for $n+1$ dimensional, $n \geq 2$, {\em simplicial} flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a …
Some curvature estimates are derived from geometrical data concerning quasi-conformality properties of some commuting linearly independent vector fields on a compact Riemannian manifold.
Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. These results generalize previous results of Liu and Terng, Pacini and Koike. In particular our results can be applied to partitions of Riemannian manifolds into orbits of actions of compact groups of isometries.
In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. We also discuss the existence of basin of attractions, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and make a few remarks on MCF of non closed leaves of generalized isoparametric foliation.
It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, bu…
A flow of metrics, $g_t$, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional $S(g)$ is a symmetric $(0,2)$-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. We introduce and study the flow of metrics on a foliation (called the 'Partial Ricci Flow'), where $S=-2 r$ and $r$ is the partial Ricci curvature of the foliation; in other words, the velocity for a unit vector $X$ orthogonal to the leaf, $-2 r(X,X)$, is the mean value of section…
We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part~II.
We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous
We prove that every closed, smooth $n$-manifold $X$ admits a Riemannian metric together with a smooth, transversely oriented CMC foliation if and only if its Euler characteristic is zero, where by CMC foliation we mean a codimension-one, transversely oriented foliation with leaves of constant mean curvature and where the value of the constant mean curvature can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of $X$ can be chosen so that the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal …
We study the geometry of the foliation by constant Gaussian curvature surfaces $(Σ_k)_k$ of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizations of the space of hyperbolic ends, defined by Labourie in 1992 in terms of the geometry of the leaves $Σ_k$. We give a new description of the renormalized volume using the constant curvature foliation. We prove a generalization of McMullen's Kleinian reciprocity theorem, which …
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 $π$ along time-like geodesics).
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique constant mean curvature hypersurface S_H in the hyperbolic n-space with asymptotic boundary C for any -1<H<1, then the constant mean curvature hypersurfaces {S_H} foliates the hyperbolic n-space.
We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov-Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.
Given a closed manifold $M$ and a vector bundle $ξ$ of rank $n$ over $M$, by gluing two copies of the disc bundle of $ξ$, we can obtain a closed manifold $D(ξ, M)$, the so-called double manifold. In this paper, we firstly prove that each sphere bundle $S_r(ξ)$ of radius $r>0$ is an isoparametric hypersurface in the total space of $ξ$ equipped with a connection metric, and for $r>0$ small enough, the induced metric of $S_r(ξ)$ has positive Ricci curvature under the additional assumptions that $M$ has a metric with positive Ricci curvature and $n\geq3$. As an application, if $M$ admits a metric …
In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass.
An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of or…
We prove that, given an acausal curve $Γ$ in the boundary at infinity of $AdS_{3}$ which is the graph of a quasi-symmetric homeomorphism $ϕ$, there exists a unique foliation of its domain of dependence $D(Γ)$ by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of $ϕ$.
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.
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.
In this paper, we prove that if a quasi-Fuchsian 3-manifold contains a minimal surface whose principle curvature is less than 1, then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow.
We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the famous theorem of Schoen-Yau and Gromov-Lawson on the non-existence of metrics of positive scalar curvature on torus to the case of foliations. Moreover, our method, which is partly inspired by the analytic localization techniques of Bismut-Lebeau, also applies to give a new proof …
The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Thus our result has applications in the theory of General Relativity.
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.
