Complete Vector Fields Research Articles

Complete polynomial vector fields on the complex plane

We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.

Zeroes of complete polynomial vector fields

We prove that a complete polynomial vector field on C 2 \mathbb {C}^{2} has at most one zero, and analyze the possible cases of those with exactly one which is not of Poincaré-Dulac type. We also obtain the possible nonzero first jet singularities of the foliation F X \mathcal {F}_X at infinity and the nongenericity of completeness. Connections with the Jacobian Conjecture are established.

Ergodicity of homogeneous Brownian flows

Let M be a finite-dimensional smooth-oriented paracompact manifold and ζ k, 0⩽k⩽d , be a family of complete smooth vector fields on M so that the Brownian flow associated with D=∑ k 1 2 ζ kζ k+ζ 0 exists globally. We prove that any volume form μ on M is irreducible for the Brownian flows if and only if there exists only constant functions ψ∈ L ∞( M, μ) satisfying the following equation: ψ=ψ∘α(ζ k,t) ∀t∈ R , 0⩽k⩽d, where (α(ζ,t) ∀t∈ R ) is the one-parameter group of diffeomorphism on M associated with the complete vector field ζ. In such a case, an invariant finite volume form μ is ergodic for the flow.

Complex vector fields having orbits with bounded geometry

Germs of holomorphic vector fields at the origin $0\in \co^{\kern1pt2}$ and polynomial vector fields on $\co^{\kern1pt2}$ are studied. Our aim is to classify these vector fields whose orbits have bounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications to the classical theory of compact foliations.

Invariant measures for Lévy flows of diffeomorphisms

We consider the Lévy flow of diffeomorphisms of a manifold obtained by solving a stochastic differential equation driven by an n-dimensional Lévy process and n complete smooth vector fields (which generate a finite-dimensional Lie algebra L) and show that a necessary and sufficient condition for a large class of such flows to have an invariant measure is that each of the vector fields is divergence-free. We investigate the existence and uniqueness of such measures and examine their invariance with respect to the action of the associated Markov semigroup. In particular, we prove that there is an invariant probability measure on M if and only if the transformation Lie group G whose Lie algebra is L is compact. In the unbounded case we show that if there is a unique G-invariant measure, then it is also the unique invariant measure for the Markov semigroup provided the flow is recurrent.

A general notion of shears, and applications.

In this paper we introduce a generalization of the notions of shears and overshears to arbitrary complex manifolds. The concept is very simple, but it is useful in the study of complex manifolds having very large automorphism groups. We shall explore some of the consequences of this concept in connection with the density property, which we now recall. In [V1] we introduced the notion of complex manifolds with the density property. Recall that a complex manifoldM has the density property if the Lie subalgebra ofXO(M) generated by the complete vector fields onM is a dense subalgebra. More generally, a Lie subalgebra g ⊂ XO(M) is said to have the density property if the complete vector fields in g generate a dense subalgebra of g. (So M has the density property if and only if XO(M) has the density property.) Another important case occurs whenM has a nonvanishing holomorphic n-form (n = dimCM), that is, a holomorphic volume element ω. We say that (M,ω) has the volume density property if the Lie algebra XO(M,ω) := {X ∈XO(M) | LXω = 0 } has the density property. Andersen [A] proved that (C, dz1 ∧ · · · ∧ dzn) has the volume density property, and then Andersen and Lempert [AL] proved thatC has the density property. The author showed that for every complex Lie groupG, (G×C, ω) has the volume density property, where ω is the unique (up to constant multiple) left (or right) invariant holomorphic volume element on G×C, and that if G is a Stein Lie group, then G × C has the density property. The author also produced several examples of Lie algebras of vector fields with the density property. In [V2] we used jets to explore the complex structure of (mostly Stein) complex manifolds with the density property. It was shown, among other things, that Stein manifolds with the density property admit open subsets biholomorphic to C and have interesting properties with respect to their embedded submanifolds. Some of the results were known for C through works of Buzzard, Fornaess, Forstneric, Globevnik, Rosay, Stensones, and others. With the usefulness of the density property already established in the literature, some sort of classification or fine structure theorem is very desirable. Such a result seems at the moment very far off, owing in part to the lack of examples. The main

Strain, displacement and rotation associated with the formation of curvature in fold belts; the example of the Jura arc

A new simplified genetic classification scheme for arcuate fold–thrust belts is proposed. Based on total strain patterns and displacement vector fields, we distinguish three extreme end-member models: (1) `Oroclines', pure bending of an initially straight belt, (2) `Piedmont glacier' with divergent transport directions and (3) `Primary arcs'. A simple geometric model set-up for the simulation of strain patterns in primary arcs with uniform transport direction demonstrates that divergent strain trajectories and rotations of passive marker lines do not require any divergence in displacement directions. These often quoted arguments are insufficient for the identification of `Oroclinal bending' or `Piedmont glacier' type of arc formation. Only three-dimensional restorations of an arc provide the critical information about displacement directions. In their absence, arc parallel stretches and rotations in comparison with total strains provide the most useful criteria for the distinction of arc formation modes. As an example, the Jura fold–thrust belt of the external Alps is discussed. A large set of strain data includes total shortening estimates based on balanced cross-sections, local strain axes orientations from the inversion of fault populations [Homberg, C., 1996. Unpublished PhD thesis, Université de Paris VI (France)], tectonic stylolites and micro-strains from twinning in sparry calcite. Strain trajectories (maximum shortening direction) computed from these data define a strongly divergent fan with a 90° opening. A complete displacement vector field for the entire Jura has been determined from balanced cross-sections augmented with three-dimensional `block mosaic' restorations [Philippe, Y., 1995. Unpublished PhD thesis, Université de Chambéry (France)]. Displacement vectors diverge by about 40°, markedly less than strain trajectories. The non-parallelism between strain trajectories and transport directions indicates that considerable wrenching deformation did occur in both limbs of the Jura arc. Paleomagnetically determined clockwise rotations of 0–13° from ten sites (Kempf, O., et al., Terra Nova 10, 6–10) behind the right-hand half of the Jura arc and two sites with a combined 23° anticlockwise rotation behind the left-hand half of the arc are and additional argument in favor of such a wrenching deformation. We conclude that the Jura arc formed as a `Primary arc' with a minor component of `Piedmont glacier' type divergence in transport directions.

Complete vector fields on (ℂ*)ⁿ

We prove necessary and sufficient conditions for a rational vector field on ( C ∗ ) n (\mathbb {C}^*)^n to be complete.

Polarization effects in near-field excitation-collection probe optical microscopy of a single quantum dot.

We solve numerically the three-dimensional vector form of Maxwell's equation for the situation of near-field excitation and collection of luminescence from a single quantum dot, using a scanning near-field optical fibre probe with subwavelength resolution. We highlight the importance of polarization-dependent effects in both the near-field excitation and collection processes. Applying a finite-difference time domain method, we calculate the complete vector fields emerging from a realistic probe structure which is in close proximity to a semiconductor surface. We model the photoluminescence from the quantum dot in terms of electric dipoles of different polarization directions, and determine the near-field luminescence images of the dot captured by the same probe. We show that a collimating effect in the high index semiconductor significantly improves the spatial resolution in the excitation-collection mode. We find that the spatial resolution, image shape and collection efficiency of near-field luminescence imaging strongly depend on the polarization direction as represented by the orientation of the radiating electric dipoles inside the quantum dot.

A full vector analysis of near-field luminescence probing of a single quantum dot

We solve numerically the three-dimensional vector form of Maxwell’s equation for the situation of near-field excitation and collection of luminescence from a single quantum dot using a scanning near-field optical fiber probe with sub-wavelength resolution. Applying a finite-difference time-domain method, we calculate the complete vector fields emerging from a realistic probe structure, as well as the near-field luminescence image of the dot captured by the same probe. We show that a collimating effect in the high index semiconductor significantly improves the spatial resolution in excitation/collection mode. We find that the spatial resolution, image shape, and collection efficiency of near-field luminescence strongly depend on the orientation of the radiating dipole in the dot.

Geometric quantization: Hilbert space structure on the space of generalized sections

The Hilbert space constructed by KS-geometric quantization consists of the solutions of the equations ▽XFs = 0 for X in the invariant complex polarization F ⊂ (TM)C. It is well known that if the holonomy groups of the restriction ▽F to the leaves of the subbundle D, DC = F ∩ F̄, are nontrivial, there are no smooth global solutions. The supports of the generalized solutions are subsets of the union BS of all D-leaves with trivial holonomy group. It is shown in this paper that for a wide class of polarizations, BS is the union of E-leaves, where EC = F + F̄. We construct a Hilbert space structure on the space of generalized sections such that any quantizable real function with a complete Hamiltonian vector field generates a one-parameter group of unitary operators.

Integration of Complex Differential Equations

Let X be a polynomial vector field in {\Bbb C}^2s then it defines an algebraic foliation {\cal F} on {\Bbb C}P(2). If {\cal F} admits a Liouvillian first integral on {\Bbb C}P(2), then it is transversely affine outside some algebraic invariant curve S\subset {\Bbb C}P(2). If, moreover, for some irreducible component S_0 \subset S, the singularities q ∈ Sing {\cal F} \cup S are generic, then either {\cal F} is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation {\cal R}: p(x)\thinspace dy-(y^2a(x)+yb(x))\thinspace dx=0 on \overline {\Bbb C} \times \overline {\Bbb C}. This result has several applications such as the study of foliations with algebraic limit sets on {\Bbb C}P(2)(2), the classification polynomial complete vector fields over {\Bbb C}^2, and topological rigidity of foliations on {\Bbb C}P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.

New examples of weakly symmetric spaces

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττk:T k+1(M)=T(T k(M))→T k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT 0(M)=M. LetV represent a possibly nonstationary vector field overT k(M), and letQ be a subset/submanifold inT k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f [k], wheref [k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail.

Classification of Killing-transversally symmetric spaces

We treat Killing-transversally symmetric spaces (briefly, KTS-spaces), that is, Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries. Such manifolds are homogeneous spaces equipped with a naturally reductive homogeneous structure and they provide a rich set of examples of reflection spaces. We prove that each simply connected reducible KTS-space $M$ is a Riemannian product of a symmetric space $M^{\prime}$ and a special kind of KTS-space $M^{\prime\prime}$, called a contact KTS-space. Such a particular manifold $M^{\prime\prime}$ is an irreducible, odd-dimensional principal $G^{1}$-bundle over a Hermitian symmetric space. The main purpose of the paper is to give a classification of this special class of manifolds $M^{\prime\prime}$.

Generation theory for semigroups of holomorphic mappingsin Banach spaces

We study nonlinear semigroups of holomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog of the Hille exponential formula. We then apply our results to the null point theory of semi-plus complete vector fields. We study the structure of null point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of the implicit function theorem and a discussion of some open problems are also included.

Locally Symmetric and Rigid Factors for Complex Manifolds via Harmonic Maps

We obtain a complete description of the Lie algebra of complete holomorphic vector fields on the universal cover of a compact complex manifold with negative first Chern class. The main tools are an equivariance result we prove for harmonic maps and the rigidity theory for harmonic maps from Kahler manifolds to locally symmetric spaces.

Limits of complete holomorphic vector fields

Recently Buzzard and Fornaess answered the question, raised in [8], as to whether there exist holomorphic vector fields on C for n ≥ 2 which can not be approximated by complete holomorphic vector fields. They showed for instance that the quadratic vector field on C, V (z, w) = z(z−1) ∂ ∂z −w ∂ ∂w , is not a limit of complete holomorphic fields; the same is true for the field z ∂ ∂z on C 2 (private communication, January 1995). Their method proves that the set of complete holomorphic vector fields is nowhere dense in the set of all entire vector fields on C, in the topology of uniform convergence on compact sets. It was clear from the results of [8] that most Hamiltonian holomorphic vector fields can not be approximated by complete Hamiltonian fields. This phenomenon is in strong contrast with the situation for smooth vector fields which can always be made complete by a modification supported outside a given compact set in M . In this article we obtain qualitative information on the fundamental domain in complex time of those holomorphic vector fields V on a Stein manifold M which are limits of complete fields (Theorem 1.1). We apply this to show that several specific types of vector fields on C are not limits of complete fields (uniformly on compacts).

Parallel integration of odes by vector field decompositions

In this paper we consider inner parallelism of ODEs, and an approach to designing large scale parallelism for integrating ODEs. It is based on vector field approximations by means of the complete integrable vector field atlas. The maps of this atlas are in agreement with the definition error of the original vector field which is changed on the local map up to some complete integrable vector field. Potential parallelism is associated with independent computations of the complete first integral system on each local map.

Isometries of non-self-adjoint operator algebras

We study isometries of certain non-self-adjoint operator algebras by means of the structure of the complete holomorphic vector fields on their unit balls and the associated partial Jordan triple products. We show that isometries of nest sub-algebras of B( H) are of the form T ↦ UTW or T ↦ UJT ∗JW , where U, W are suitable unitary operators and J a fixed involution of H.