Classes Of Foliations Research Articles

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.

Classification of foliations of degree three on ℙ ℂ 2 with a flat Legendre transform

Classification of foliations of degree three on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>ℙ</mml:mi> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup></mml:math> with a flat Legendre transform

Toward Effective Liouvillian Integration

We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension.

Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations

Abstract Let C be a smooth elliptic curve embedded in a smooth complex surface X such that C is a leaf of a suitable holomorphic foliation of X. We investigate the complex analytic properties of a neighborhood of C under some assumptions on the complex dynamical properties of the holonomy function. As an application, we give an example of ( C , X ) {(C,X)} in which the line bundle [ C ] {[C]} is formally flat along C, however it does not admit a C ∞ {C^{\infty}} Hermitian metric with semi-positive curvature. We also exhibit a family of embeddings of a fixed elliptic curve for which the positivity of normal bundles does not behave in a simple way.

Analytic Classification of Foliations Induced by Germs of Holomorphic Vector Fields in ( ℂ n , 0 ) $(\mathbb {C}^{n},0)$ with Non-isolated Singularities

We consider germs of holomorphic vector fields in $(\mathbb {C}^{n},0)$ , n ≥ 3, with non-isolated singularities. We assume that the set of singular points forms a submanifold of codimension 2, and the sum of the nonzero eigenvalues of the linearization of the germs at each singular point is zero. We give the orbital analytic classification of generic germs of such type. It happens that, unlike the formal classification (which is trivial), the analytic one has functional moduli. The same result is obtained in the real-analytic case (the smooth normalization was obtained earlier in Roussarie Asterisque. 1975;30:1–181).

Homotopy classification of contact foliations on open contact manifolds

We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontact immersions on open contact manifolds.

Classification of Foliations on CP^2 of degree 3 with Degenerate Singularities

Classification of Foliations on CP^2 of degree 3 with Degenerate Singularities

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

One can easily give examples of rank $2$ flat connections over $\mathbb{P}^2$ by rational pull-back of connections over $\mathbb{P}^1$. We give an example of a connection that can not occur in this way; this example is constructed from an algebraic solution of Painleve VI equation. From this example we deduce a Hilbert modular foliation. The proof of this relies on the classification of foliations on projective surfaces due to Brunella, Mc Quillan and Mendes. Then, we get the dual foliation and, by a precise monodromy analysis, we see that our twice foliated surface is covered by the classical Hilbert modular surface constructed from the action of $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ on the bidisc.

Codimension-one foliations calibrated by nondegenerate closed 2-forms

A class of codimension-one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally nondegenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson-type submanifold. In particular we deduce that their leaf spaces are homeomorphic to leaf spaces of 3-dimensional taut foliations. We also introduce surgery constructions to show that this class of foliations is broad enough. Our techniques come mainly from symplectic geometry.

The K-theory of the C*-algebra of foliations by slope components

AbstractWe compute the K-theory of the C*-algebra for a large class of foliations of the 3-torus, which contains in particular all smooth foliated circle bundles over the 2-torus. This generalizes a well-known result of Torpe. We show that the rank of the K-theory groups reflect part of the geometrical aspect of the foliation. To illustrate these results, we compute some concrete examples, including a case where both K-theory groups have infinite rank.

Hyperfoliations on compact 3-manifolds with restrictions on the external curvature of leaves

C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic, elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS3 such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional curvature of constant sign admitting parabolic foliations are classified.

Lorentzian approach to black hole thermodynamics in the Hamiltonian formulation

In this work, we extend the analysis of Brown and York to find the quasilocal energy in a spherical box in the Schwarzschild spacetime. Quasilocal energy is the value of the Hamiltonian that generates unit magnitude proper-time translations on the box orthogonal to the spatial hypersurfaces foliating the Schwarzschild spacetime. We call this Hamiltonian the Brown-York Hamiltonian. We find different classes of foliations that correspond to time-evolution by the Brown-York Hamiltonian. We show that although the Brown-York expression for the quasilocal energy is correct, one needs to supplement their derivation with an extra set of boundary conditions on the interior end of the spatial hypersurfaces inside the hole in order to obtain it from an action principle. Replacing this set of boundary conditions with another set yields the Louko-Whiting Hamiltonian, which corresponds to time-evolution of spatial hypersurfaces in a different foliation of the Schwarzschild spacetime. We argue that in the thermodynamical picture, the Brown-York Hamiltonian corresponds to the internal energy whereas the Louko-Whiting Hamiltonian corresponds to the Helmholtz free energy of the system. Unlike what has been the usual route to black hole thermodynamics in the past, this observation immediately allows us to obtain the partition function of such a system without resorting to any kind of Euclideanization of either the Hamiltonian or the action. In the process, we obtain some interesting insights into the geometrical nature of black hole thermodynamics.

Non-uniquely ergodic foliations of thin type

We construct a minimal foliation of thin type which is not uniquely ergodic. The notion of thin type relates to Rips' classification of foliations on 2-complexes.

Stability for a class of foliations covered by a product.

Stability for a class of foliations covered by a product.

Meromorphic singular foliations on complex projective surfaces

Here we show that many numerical computations and bounds on the degrees of the algebraic leaves for singular meromorphic foliations on ℂℙ2 may be extended to large classes of foliations and complex projective surfaces.

Small commutators of piecewise linear homeomorphisms of the real line

IN THIS PAPER we study the group P&(R) of piecewise linear (PL) homeomorphisms of the real line R with compact support. This group has been studied by many people. As for the homology of this group, Epstein showed that the group is a perfect group and hence it is a simple group [2]. The lower-dimensional homology of this group was determined by Greenberg [S]. In particular, his result says that the 2-dimensional homology is R @Jz R and it is easy to see that the canonical bilinear map R Oz R + R is nothing but the discrete Godbillon-Vey class described in [6] and [4]. In [22], we determined all the homology group using Greenberg’s description [8] of the classifying space for transversely PL foliations. Our study on the group PL,(R) was motivated by an intention to use these results on the homology of PL,(R) to understand the homology of groups of Lipschitz homeomorphisms of R. In particular, since the (discrete) Godbillon-Vey 2-cocycle for PL,(R) was completely understood, we tried to understand the Godbillon-Vey 2-cocycle for groups of Lipschitz homeomorphisms of R using approximations by elements of the group PL,(R). The Godbillon-Vey invariant was first defined for codimension-1 C2 foliations of closed oriented 3-manifolds [7]. This invariant is the only known non-trivial invariant for C2 foliated cobordism and it varies continuously under the deformation of the foliation [ 141. For transversely oriented codimension-1 C’ foliations, the classifying space for them is contractible [ 1 S] and such foliations of closed oriented 3-manifolds are all cobordant. This fact has already been shown for transversely oriented Lipschitz foliations [l 11. Hence the Godbillon-Vey invariant cannot be defined for codimension-1 C’ or Lipschitz foliations. There are, however, a lot of classes of foliations between C’ or Lipschitz and C2. In fact, the Godbillon-Vey invariant was extended to the foliations of class C’ +a (CI > f) by Hurder and Katok [9], and to the transversely piecewise linear (or piecewise C*) by Ghys and Sergiescu [6,4]. We defined the Godbillon-Vey invariant for the foliations of class CLvyn with (1 I fl l/2) and the transversely PL foliations. Since each cobordism class of the foliations of class CL’% (1 5 j? < 2) of closed oriented 3-manifolds contains a representative which is a 2-cycle for the group GfvYn(R) of diffeomorphisms of class C ‘,% of the real line R with compact support [12], the key problem is to understand the Godbillon-Vey invariant on 2-cycles of the group G:*“(R). For a real number j3 (p 2 l), the group G, L*vn(R) of CL’Yn diffeomorphisms of R with compact support is defined as follows. An element fof G, L*Y#(R) is a Lipschitz homeomorphism with compact support such that logf’(x 0) is with bounded p-variation (see

Admissibility conditions for symbolic sequences of the Lozi map

In terms of contracting and expanding foliations the two-dimensional Lozi map is decomposed into two coupled one-dimensional maps. Symbolic sequences assigned to the two classes of foliations are well ordered. The pruning front is constructed from tangencies between the two classes of foliations. A necessary condition and a sufficient condition for admissible sequences are proposed.

Symbolic Dynamics for the Tél Map11The project partially supported by National Natural Science Foundation of China.

In terms of contracting and expanding foliations the two-dimensional Tél map is decomposed into two coupled one-dimensional maps. Symbolic sequences are assigned to the two classes of foliations, and their ordering is discussed. The pruning front is constructed. A necessary and sufficient condition for admissible sequences is proposed.

La conjecture de Baum-Connes pour un feuilletage sans holonomie de codimension un sur une variété fermée

In [C2], Baum-Connes state a conjecture for the K-theory of C*-algebras of foliations. This conjecture has been proved by T. Natsume [N2] for C8-codimension one foliations without holonomy on a closed manifold. We propose here another proof of the conjecture for this class of foliations, more geometric and based on the existence of the Thom isomorphism, proved by A. Connes in [C3]. The advantage of this approach is that the result will be valid for all C0-foliations.

The dynamics of open, foliated manifolds and a vanishing theorem for the Godbillon-Vey class

Let (M,.F) be a C3-foliated n-manifold with leaves of dimension n 1. Let gv(F) E H3(M, IR) denote the characeristic class introduced by Godbillon and Vey in [G-V]. Considerable effort has been made to discover the geometric significance of this invariant (recall Thurston’s “helical wobble” [Th], which can be interpreted via the formulas in [R-W]). In addition to celebrated examples of foliations for which gv(F) does not vanish [G-V, Th, Br], the list of foliations for which it does has been growing steadily. By work of Herman [Her] and Wallet [WI, gv(Sr) = 0 for foliated circle bundles and interval bundles over T2. This result has interesting consequences that are far from being obvious. Thus, Morita and Tsuboi have deduced the vanishing of gv(X) for closed, foliated manifolds without holonomy [M-T] and Mizutani, Morita, and Tsuboi have extended this to foliations almost without holonomy [M-M-T]. In all of the above vanishing theorems, the leaves of the foliations have polynomial growth. Other classes of foliations, with all leaves polynomial and gv(,F) = 0, were found by Nishimori [N], Sergiescu [Ser], and Tsuchiya [Ts]. In a preprint [C-C2], that became obsolete within a few months of its distribution, the present authors proved the vanishing theorem for all closed, foliated manifolds in which every leaf has polynomial growth. Indeed, we proved that, if every leaf is nonresilient [L] and if no leaf lies at infinite level, then gv(sT) = 0.