The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $β$ defining a foliation on a branched simple polyhedron $P$ satisfies $dβ>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $β$ on $P$ with $dβ>0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $β$ with $dβ>0$ on the abalone.
We study the group of leafwise holomorphic smooth automorphisms of Reeb components of leafwise complex foliation which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we determine the structure of the group of leafwise holomorphic automorphisms.
We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.
We realize the Reeb foliation of S^3 as a family of Legendrian submanifolds of the unit S^5 \subset C^3, Moreover we construct a deformation of the standard contact S^3 in S^5, via a family of contact submanifolds, into this realization.
This article describes the following results which relate to each other; i) convergence of high dimensional contact structure to codimension one foliation with Reeb component, ii) relation between Nil-type and Sol-type contact submanifolds of S^5, iii) definition of convex Thurston-Bennequin inequality, and iv) generalization of Lutz twist via convex hypersurface theory. The other article concerning non-convex hypersurfaces is included as an appendix.
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.
The original article titled "Variétés (non séparées) à une dimension et structures feuilletées du plan" was published in 1957 in French in L'Enseignement Mathématique. It establishes a beautiful connection between foliations of the plane and non-Hausdorff $1$-dimensional manifolds arising naturally as leaf spaces of the foliations. Since its appearance, this theory has paved the way for several results concerning dynamical systems and foliations of the plane and $2$-manifolds. Haefliger and Reeb's article inspires many results in the theory of foliation of $3$-manifolds as well: we refer the i…
The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components obtained by the Hopf construction, we have an almost complete description of the groups of leafwise holomorphic smooth automorphisms.
We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in…
We introduce a notion of positive pair of contact structures on a 3-manifold which generalizes a previous definition of Eliashberg-Thurston and Mitsumatsu. Such a pair gives rise to a locally integrable plane field $λ$. We prove that if $λ$ is uniquely integrable and if both structures of the pair are tight, then the integral foliation of $λ$ doesn't contain any Reeb component whose core curve is homologous to zero. Moreover, the ambient manifold carries a Reebless foliation. We also show a stability theorem "à la Reeb" for positive pairs of tight contact structures.
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…
On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to a taut foliation. By appealing to results of Hofer, Wysocki, and Zehnder, we show that, under certain conditions, the 3-manifold obtained by Dehn filling along bdry V is irreducible and different from the 3-sphere. Resumé On construit, sur toute variete V de dimension trois orien…
Let $M$ be a closed orientable irreducible $3$-manifold such that $π_1(M)$ is left orderable. (a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $(π_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,Γ)$ composed of a left-invariant order "$<$" of $π_1(M)$ and a fundamental domain $Γ$ of…
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…
Morse theory and moduli spaces of simple polylines and polygons
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；School of Mathematics, Hunan University（湖南大学数学学院）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）
