One-dimensional Foliation Research Articles

Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps

We construct holomorphic support functions for smooth weakly pseudoconvex hypersurfaces with Levi form of constant rank. These are then applied to show that formal holomorphic curves which are tangential to infinite order to such a hypersurface must be formally contained in its Levi foliation. As a consequence, we obtain a holomorphic deformation theorem for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, strengthening a very general result of Lamel and Mir about formal deformations in this particular case.

Global residue formula for logarithmic indices of one-dimensional foliations

Global residue formula for logarithmic indices of one-dimensional foliations

Leafwise quasigeodesic foliations in dimension three and the funnel property

AbstractWe construct one-dimensional foliations which are subfoliations of two-dimensional foliations in$3$-manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R×(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 R×t, t∊(0,1), and boundary intervals which gives a foliation Δ on all of Z. Denote by H(Z,Δ) the group of all homeomorphisms of Z that maps leaves of Δ onto leaves and by H(Z/Δ) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π0H(Z,Δ) with a group of automorphisms of a certain graph G with additional structure which encodes the combinatorics of gluing Z from strips. That graph is in a certain sense dual to the space of leaves Z/Δ.
 On the other hand, for every h\inH(Z,Δ) the induced permutation k of leaves of Δ is in fact a homeomorphism of Z/Δ and the correspondence h→k is a homomorphism ψ:H(Δ)→H(Z/Δ). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ0:π0H(Z,Δ)→π0H(Z/Δ) which turns out to be either injective or having a kernel Z2. This gives a dual description of π0H(Z,Δ) in terms of the space of leaves.

The nuclear dimension of C⁎-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow R↷Y, the resulting crossed product C⁎-algebra C0(Y)⋊R has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of C⁎-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of Z. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation C⁎-algebras and crossed products.

On the resolution of singularities of one-dimensional foliations on three-manifolds

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.

Endperiodic automorphisms of surfaces and foliations

We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations${\mathcal{F}}$and${\mathcal{F}}^{\prime }$are transverse to a common one-dimensional foliation${\mathcal{L}}$whose monodromy on the non-compact leaves of${\mathcal{F}}$exhibits the nice dynamics of Handel–Miller theory, then${\mathcal{L}}$also induces monodromy on the non-compact leaves of${\mathcal{F}}^{\prime }$exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.

Homeotopy groups of one-dimensional foliations on surfaces

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 $\Delta$ 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 $\mathbb{R}^2 \to \mathbb{R}$ without singularities. Previously, the first two authors studied the homotopy type of the group $\mathcal{H}(\Delta)$ of homeomorphisms of $Z$ sending leaves of $\Delta$ onto leaves, and shown that except for two cases the identity path component $\mathcal{H}_{0}(\Delta)$ of $\mathcal{H}(\Delta)$ is contractible.The aim of the present paper is to show that the quotient $\mathcal{H}(\Delta)/ \mathcal{H}_{0}(\Delta)$ can be identified with the group of automorphisms of a certain graph with additional structure encoding the ``combinatorics'' of gluing.

Solvable groups and a shear construction

The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geometric structures that may be obtained from the shear construction.

Pfaffian intersections and multiplicity cycles

We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point p by the local algebraic multiplicity at p of a suitably constructed algebraic cycle. The construction is based on Gabrielov’s complex analog of the Rolle–Khovanskiĭ lemma. We illustrate the main result by deriving similar uniform estimates for the complexity of the Milnor fiber of a deformation (under a smoothness assumption) and for the order of contact between an algebraic hypersurface and an arbitrary non-singular one-dimensional foliation. We also use the main result to give an alternative geometric proof for a classical multiplicity estimate in the context of commutative group varieties.

Invariant sets near singularities of holomorphic foliations

Consider a complex one-dimensional foliation on a complex surface near a singularity $p$. If ${\mathcal{I}}$ is a closed invariant set containing the singularity $p$, then ${\mathcal{I}}$ contains either a separatrix at $p$ or an invariant real three-dimensional manifold singular at $p$.

Foliations singular along a curve

A general one-dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes in ℙ n , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d. We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive-dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.

A Darboux-type theorem for germs of holomorphic one-dimensional foliations

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

Foliated neighborhoods of exceptional manifolds

The present article deals with the classification of neighborhoods of negatively embedded submanifoldsA of a com- plex manifold X. The main tools we use are one-dimensional foliations whose set of singularities is A and which are normally attracting at A. The linearization of these foliations is provided under general cohomological conditions. As a consequence, an ex- tension of the classical embedding theorem of Grauert is obtained.

Foliations by curves with curves as singularities

Soit ℱ un feuilletage holomorphe unidimensionnel sur ℙ n , dont les composantes du lieu singulier Σ sont des courbes C i et des points p j . On exprime le nombre de tels points p j , comptés avec leurs multiplicités, en termes des invariants de ℱ et C i , en supposant que ℱ est spécial le long des courbes C i . En supposant qu’il n’y a qu’une seule composante de Σ de dimension non nulle, on obtient aussi des résultats lorsque le feuilletage est déterminé par ses lieux singuliers.

Almost periodic structures and the semiconjugacy problem

The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80ʼs. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost periodic base flows, mathematical quasicrystals or maps of the real line with almost periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows the treatment of scalar differential equations that are almost periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds.

Invariants of transverse foliations

We construct two invariants for a pair of transverse one-dimensional foliations on the plane. If the set of separatrices is Hausdorff in the space of leaves, the invariant is a distinguished graph. In case there are a finite number of separatrices the invariant is an indexed link.

A Poincaré type inequality for one-dimensional multiprojective foliations

We study one-dimensional holomorphic foliations on products of complex projective spaces and present results giving the number of singularities, counting multiplicities, of a generic foliation, a criterion for a foliation to be Riccati and a Poincare type inequality, relating degrees of foliations to degrees of hypersurfaces which are invariant by them.

ONE-DIMENSIONAL LIE FOLIATIONS WITH GENERIC SINGULARITIES IN COMPLEX DIMENSION THREE

AbstractWe prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

Contact geometry of one-dimensional complex foliations

Contact geometry of one-dimensional complex foliations