Nonsingular Vector Fields Research Articles

On the rigidity of mean curvature flow solitons in certain semi-Riemannian warped products

We obtain rigidity results concerning complete noncompact solitons of the mean curvature flow related to a nonsingular Killing vector field $K$ globally defined in a semi-Riemannian space, which can be modeled as a warped product whose base corresponds to a fixed integral leaf of the distribution orthogonal to $K$ and the warping function is equal to $|K|$. Our approach is based on a suitable maximum principle dealing with a notion of convergence to zero at infinity. As application, we study the uniqueness of solutions for the mean curvature flow soliton equation in these ambient spaces.

Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity

Assuming suitable constraints on the warping function ρ and on the mean curvature vector field, we apply the Omori-Yau's generalized maximum principle in order to study the behavior of a support function naturally attached to a complete n-dimensional submanifold Σn immersed in a Killing warped product Mn+p×ρR whose base Mn+p has sectional curvature bounded from below. In the case that Σn is compact, we prove the nonexistence of non-minimal compact mean curvature flow solitons with respect to a non-singular Killing vector field. When Σn has codimension 2, we conclude that there do not exist n-dimensional minimal compact submanifolds immersed in Mn+1×ρR whose base is either Mn+1=Rn+1 or Mn+1=Hn+1. We also get that the only minimal topological 2-spheres immersed in S3×ρR are the totally geodesic ones in a slice S3×{t0}. Moreover, we obtain further results concerning the geometry of complete submanifolds immersed in a Killing warped product, under appropriate restrictions on the Bakry-Émery-Ricci tensor, Ricψ, of these submanifolds with respect to the weight function ψ=ln⁡ρ−2.

On the dynamics of some vector fields tangent to non-integrable plane fields

Let $\mathcal{E}^3\subset TM^n$ be a smooth $3$-distribution on a smooth manifold of dimension $n$ and $\mathcal{W}\subset\mathcal{E}$ a line field such that $[\mathcal{W},\mathcal{E}]\subset\mathcal{E}$. Under some orientability hypothesis, we give a necessary condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W}\subset\mathcal{D}$ and $[\mathcal{D},\mathcal{D}]=\mathcal{E}$. Moreover we study the case where a section of $\mathcal{W}$ is non-singular Morse-Smale and we get a sufficient condition for the global existence of $\mathcal{D}$. As a corollary we get conditions for a non-singular vector field $W$ on a $3$-manifold to be Legendrian for a contact structure $\mathcal{D}$. Similarly with these techniques we can study when an even contact structure $\mathcal{E}\subset TM^4$ is induced by an Engel structure $\mathcal{D}$.

Topological transport of vorticity in Heisenberg magnets

We study a robust topological transport carried by vortices in a thin film of an easy-plane ferromagnetic insulator between two metal contacts. A vortex, which is a nonlocal topological spin texture in two-dimensional magnets, exhibits some beneficial features as compared to skyrmions, which are local topological defects. In particular, the total topological charge carried by vorticity is robust against local fluctuations of the spin order-parameter magnitude. We show that an electric current in one of the magnetized metal contacts can pump vortices into the insulating bulk. Diffusion and nonlocal Coulomb-like interaction between these vortices will establish a steady-state vortex flow. Vortices leaving the bulk produce an electromotive force at another contact, which is related to the current-induced vorticity pumping by the Onsager reciprocity. The voltage signal decays algebraically with the separation between two contacts, similarly to a superfluid spin transport. Finally, the vorticity and closely related skyrmion type topological hydrodynamics are generalized to arbitrary dimensions, in terms of nonsingular order-parameter vector fields.

Extended Fuller index, sky catastrophes and the Seifert conjecture

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

<inline-formula><tex-math id="M1">\begin{document}$C^1$\end{document}</tex-math></inline-formula> weak Palis conjecture for nonsingular flows

This paper focuses on generic properties of continuous dynamical systems. We prove \begin{document}$C^1$\end{document} weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among \begin{document}$C^1$\end{document} nonsingular vector fields. Our arguments contain three main ingredients: linear Poincare flow, Liao's selecting lemma and the adapting of Crovisier's central model. Firstly, by studying the linear Poincare flow, we prove for a \begin{document}$C^1$\end{document} generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincare flow. Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.

Foliated vector fields without periodic orbits

In this article parametric versions of Wilson’s plug and Kuperberg’s plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and its subspace comprised of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds.

Multiplicity estimates: A Morse-theoretic approach

The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a non-singular polynomial vector field, at one or several points, has been considered by authors in several different fields. The two best (incomparable) estimates are due to Gabrielov and Nesterenko. In this paper we present a refinement of Gabrielov's method which simultaneously improves these two estimates. Moreover, we give a geometric description of the multiplicity function in terms certain naturally associated polar varieties, giving a topological explanation for an asymptotic phenomenon that was previously obtained by elimination theoretic methods in the works of Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton polytopes, strongly generalizing the classical estimates.

A Linking/S1-equivariant Variational Argument in the Space of Dual Legendrian Curves and the Proof of the Weinstein Conjecture on S3 ”in the Large”

Abstract Let α be a contact form on S3, let ξ be its Reeb vector-field and let v be a non-singular vector-field in ker α. Let Cβ be the space of curves x on S3 such ẋ = aξ + bv, ȧ = 0, a ≩ 0. Let L+, respectively L−, be the set of curves in Cβ such that b ≥ 0, respectively b ≤ 0. Let, for x ∈ Cβ, J(x) = ∫0 1 αx(ẋ)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1-equivariant homology of Cβ, relative to L+ ∪ L− and to some specially designed ”bottom set”, see section 4, are achieved in the Morse complex of (J,Cβ) by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of ξ; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k−1) = 1+(2k−2), 1 for the linking, (2k−2) for the S1-equivariance, we find that the equivariant contributions of a critical point at infinity to L+ and to L− are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for ξ turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the ”bottom sets” of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that πn+1(Sn) = Z2, n ≥ 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of ξ excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L+ and L−. It therefore follows from the present work that either a periodic orbit of index 1 connects L+ and L− (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of ξ, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.

A note on homotopy classes of nonsingular vector fields on [formula omitted

The homotopy class (up to homeomorphism) of nonsingular vector fields on S3 are in one-to-one correspondence with N via the homotopy number. We prove that each homotopy class with a nonzero homotopy number can be represented by two nonsingular Morse–Smale vector fields with three periodic orbits. Notice that it is already known that the nonsingular Morse–Smale vector field with two periodic orbits has homotopy number 0.

The Reidemeister-Turaev torsion of standard Spin c structures on Seifert fibered 3-manifolds

The Reidemeister-Turaev torsion is an invariant of 3-manifolds equipped with Spin c structures. Here, a Spin c structure of a 3-manifold is a homology class of non-singular vector fields on it. Each Seifert fibered 3-manifold has a standard Spin c structure, which is represented as a non-singular vector field the set of whose orbits give a Seifert fibration. We provide an algorithm for computing the Reidemeister-Turaev torsion of the standard Spin c structure on a Seifert fibered 3-manifold. The machinery used to compute the torsion is that of punctured Heegaard diagrams.

ENERGY OF GLOBAL FRAMES

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

Complex Polynomial Vector Fields Having a Finitely Curved Orbit

The author proves that a non-singular polynomial vector field without invariant lines and having an entire finitely curved transcendent orbit on ℂ2 must be equivalent to a trivial vector field by a holomorphic change of coordinates. Other classification results are obtained for polynomial vector fields having a finitely curved orbit on ℂ2.

About homotopy classes of non-singular vector fields on the three-sphere

Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano: every non-singular vector field on the three-sphere is homotopic to a non-singular Morse-Smale vector field.

Reidemeister-Turaev torsion of 3-dimensional�Euler structures with simple boundary tangency�and pseudo-Legendrian knots

We generalize Turaev's definition of torsion invariants of pairs (M,�), where M is a 3-dimensional manifold andis an Euler structure on M (a non-singular vector field up to homotopy relative to @M and local modifications in Int(M)). Namely, we allow M to have arbitrary boundary andto have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's H1(M)-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) pseudo- Legendrian knots (i.e. knots transversal to a given vector field), and hence to Legendrian knots in contact 3-manifolds. We show that torsion, as an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic. Using branched standard spines to describe vector fields we show how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation of torsions a more effective one. As an example we work out a specific calculation.

Kerr and Schwarzschild black-holes in three-dimensional relativity

We consider the Kerr and Schwarzschild black-hole space–times in the framework of a three-dimensional formulation of relativistic kinematics and field dynamics, in which local physical observers are represented by non-singular vector fields of bounded length in a three-dimensional pseudo-Riemannian space. A space–time is represented by a pair of 3-metric and a fundamental 3-vector field satisfying a set of basic equations, each solution of which determines uniquely a solution of the vacuum Einstein field equations. It is shown that the only spherically symmetric solution of our basic equations leads to the Schwarzschild space–time, thereby proving a version of Birkoff’s theorem in this formalism. The Schwarzschild horizon and the Kerr stationary limit are both related to the upper bound of the length of the corresponding physical 3-vector fields. An example of a solution of the vacuum Einstein field equations shows that nonstationary space–times can also be formulated in this three-dimensional relativity.

On pairs of foliations defined by vector fields in the plane

We obtain a smooth and analytic local classification of pairs of foliations of theplane assuming that one of the foliations is defined by a nonsingular vector field and theother foliation is defined by a singular vector field.

Nonsingular vector fields in ð1(M3) satisfy Axiom A and no cycle: a new proof of Liao's theorem

Nonsingular vector fields in ð1(M3) satisfy Axiom A and no cycle: a new proof of Liao's theorem

On the complex geometry of a class of non-Kählerian manifolds

In a recent paper a class of complex, compact and non-Kahlerian manifolds was constructed by S. Lopez de Medrano and A. Verjowsky. This class contains as particular cases Calabi-Eckmann manifolds, almost all Hopf manifolds and many of the examples given previously by J.-J. Loeb and M. Nicolau. In this paper we show that these manifolds are endowed with a natural non-singular vector field which is transversely Kahlerian, and that analytic subsets of appropriate dimensions are tangent to this vector field. This permits to give a precise description of these sets in the generic case. In the proof, an important role is played by some complex abelian groups which are biholomorphic to big domains in these manifolds.

Singular relative integral invariants and adiabatic processes of thermodynamics

UDC 517.987.5; 517.958 The vector field is rectified in the neighborhood of a nonsingular point. By means of local foliation into nonsingular hypersurfaces containing phase curves of the vector field, we obtain a two-dimensional vector quotient field. A quotient field in the phase plane defines the rectified phase pattern and the module (over the ring of first integrals) of invariant measures. The pair that is composed of the phase pattern and the invariant measure of the vector field in the plane has a stationary subaigebra in the algebra of all vector fields. The triple consisting of the initial vector field, its invariant measure, and an element of the stationary subalgebra of the pairs, admits a relative integral invariant, namely, the integral of the measure over sets that are bounded by the phase curves of two vector fields. The relative integral invariant is generated (over the ring of first integrals) by the action of an element of the stationary subalgebra (the product of the Hamiltonian or the potential by the time of the element of the subalgebra). This explicitly asymmetric construction relative to the pair of vector fields becomes symmetric if we consider singular invariant measures and smooth vector fields. Suppose that an element of the subalgebra is a nonsingular vector field at the point being considered. Then we can take it as the initial element for the next triple. Obviously, the iteration of this procedure generates a chain of triples and, respectively, a chain of measures. We are interested in the convergence of the arising measures (the sense of convergence is defined in the article). We say that a chain of triples is subjected to the weight k > 0 (say, k is a positive integer) if, in each triple, the invariant measure generated by the action of the element of the subaigebra is generated by the first integral of the first element of the triple of degree k (The invariant measure of a triple makes it possible to determine two numbers, namely, the successive weight, i.e., the order of singularity of the measure generated by the action of the element of the subalgebra over the ring of the first integrals of the subalgebra and the preceding weight, which is the same as above, only with the replacement of the element of the subalgebra by the initial vector field.). The order of singularity of an invariant measure is the weight of the measure relative to the action for the element of the subalgebra. Theorem. The orders of singularity of the invariant measures of a chain of triples subjected to the weight k correspond to the orbits of the Feigenbaum mapping.