Let F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of arbitrary codimension if there exists a closed p form which evaluates positively on every compact leaf. For foliations of codimension 1 on compact manifolds it is known that the union of all non-compact leaves is an open set [A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367-397].
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
This article proves that the parity of the number of Klein-bottle leaves in a smooth cooriented taut foliation is invariant under smooth deformations within taut foliations, provided that every Klein-bottle leaf involved in the counting has non-trivial linear holonomy.
We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows faster than any recursive function.
Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some non-negative integer $k$, then the foliation $\mathcal{F}$ is without holonomy.
Let $N$ be a compact manifold with a foliation $\mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $\mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,\mathscr{F}_N) \rightarrow (M,\mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,\mathscr{F}_N)$ and $h(M,\mathscr{F}_M)$ and it holds $h(M,\mathscr{F}_M)…
Torus leaves play a crucial role in the theory of foliations. For example non-taut foliations admit a torus leaf (see the article of Goodman). In this paper, we study all the foliations near a torus leaf, and try to understand why sometimes it is taut, or non-taut (and Reebless). We focus on some crucial examples to understand non-taut and Reebless foliations; and foliations admitting a non-separating torus leaf. It relies on the study of turbulization and spiraling, with generalizations.
A transitive compact foliated space is shown to be a Riemannian foliation if and only if it is locally connected, finite dimensional, strongly equicontinuous and quasi-analytic, and the closure of its holonomy pseudogroup is quasi-analytic.
We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theorem, part of Epstein's local structure theorem for foliations by compact leaves, and a continuity theorem of Alvarez and Candel. Several examples are discussed.
We present new open manifolds that are not homeomorphic to leaves of any C^0 codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension 5 or greater that are non-periodic in homotopy or homology, namely in their 2-dimensional homotopy or homology groups.
Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations…
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is …
We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question of theirs concerning bounds on self-intersection numbers of multisections in surface bundles.
We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.
We construct a smooth codimension-one foliation on the five-sphere in which every leaf is a symplectic four-manifold and such that the symplectic structure varies smoothly. Our construction implies the existence of a complete regular Poisson structure on the five-sphere.
Every open manifold L of dimension greater than one has complete Riemannian metrics g with bounded geometry such that (L,g) is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of (L,g) suffice to make it quasi-isometric to a leaf of such a foliation. We introduce the `bounded homology property', a semi-local property of (L,g) that is necessary for it to be a leaf in a compact manifold in codimension one, up to quasi-isometry. An essential step involves a partial generalization of the Novikov closed leaf theorem to high…
A foliation of a manifold M is called R-covered if its lift to the universal cover of M has space of leaves R. We show that there are many graph manifolds which admit taut foliations, but which do not admit any R-covered foliations. On the other hand, we show that these manifolds all have finite covers admitting R-covered foliations.
Let $f:M\to M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. One important corollary is that if F is a Reebless finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classe…
We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^{2}$ foliation on a compact manifold. These examples include some exotic ${\mathbb R}^4$'s and exotic cylinders $S^3\times{\mathbb R}$.
A singular riemannian foliation on a complete riemannian manifold is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such singular foliation is the partition by orbits of a polar action,e.g. the orbits of the adjoint action of a compact Lie group on itself. We pro…
Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.
A taut foliation of a hyperbolic 3-manifold has the continuous extension property for leaves in almost every direction; that is, for each leaf of the universal cover of the foliation and almost every geodesic ray in the leaf, the limit of the ray in the universal cover of the 3-manifold is a well-defined point in the ideal boundary.
For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carrière. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.
We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.
Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.
Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $π_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $π_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter.
In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidian space. This result gives us local information about the singular foliation and in particular about the singular stratification of the foliation. It also allows us to describe the plaques of the foliation as level sets of a t…
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $Δ$ be a foliation on $X$ such that each leaf $ω\inΔ$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cuttin…
Topology of singular foliations of closed 1-forms on orbifolds
Instituto de Matemática e Estatística, Universidade Federal Fluminense（联邦弗拉明戈大学数学与统计研究所）；Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计研究所）
Let $X$ be an $(n+1)$-dimensional manifold, $Δ$ be a one-dimensional foliation on $X$, and $p: X \to X / Δ$ be a quotient map. We will say that a leaf $ω$ of $Δ$ is special whenever the space of leaves $X / Δ$ is not Hausdorff at $ω$. We present necessary and sufficient conditions for the map $p: X \to X / Δ$ to be a locally trivial fibration under assumptions that all leaves of $Δ$ are non-compact and the family of all special leaves of $Δ$ is locally finite.
Let $Δ$ be a foliation on a topological manifold $X$, $Y$ be the space of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the factor topology with respect to $p$. Then the group $\mathcal{H}(X, Δ)$ of foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts on the space of leaves $Y$, which gives a homomorphism $ψ: \mathcal{H}(X, Δ) \to \mathcal{H}(Y)$. We present sufficient conditions when $ψ$ is continuous with respect to the corresponding compact open topologies. In fact similar results hold not only for foliations but for a more general class of pa…
Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals which gives a foliation $Δ$ on all of $Z$. Denote by $\mathcal{H}(Z,Δ)$ the group of all homeomorphisms of $Z$ that maps leaves of $Δ$ onto leaves and by $\mathcal{H}(Z/Δ)$ the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologi…
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise small…
Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $Δ$ on all of $Z$. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "striped" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{…
Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described for homogeneous foliations with dense leaves on closed nilmanifolds.
