Volume-preserving Vector Fields Research Articles

Limiting Measures and Energy Growth for Sequences of Solutions to Taubes’s Seiberg–Witten Equations

We consider sequences of solutions (psi _n,A_n)_{n=1}^infty to Taubes’s modified Seiberg–Witten equations, associated with a fixed volume-preserving vector field X on a 3-manifold and corresponding to arbitrarily large values of the strength parameter r_nrightarrow infty . In Taubes’s work, the asymptotic behavior of these solutions is related to the dynamics of X. We consider the rather unexplored case of sequences of solutions whose energy is not uniformly bounded as nrightarrow infty . Our first main result shows that when the energy grows more slowly than r_n^{1/2}, the limiting nodal set of the solutions converges to an invariant set of the vector field X. The main tool we use is a novel maximum principle for the solutions with the key property that it remains valid in the unbounded energy case. As a byproduct, in the usual case of sequences of solutions with bounded energy, we obtain a new, more straightforward proof of Taubes’s result on the existence of periodic orbits that does not involve a local analysis or the vortex equations. Our second main result proves that, contrary to what happens in the bounded energy case, when the energy is unbounded there are no local restrictions to the limiting measures that may arise in the modified Seiberg–Witten equations. Furthermore, we obtain a connection between the dimension of the support of the limiting measure (as expressed through a d-Frostman property) and the energy growth of the sequence of local solutions we construct.

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

Volume preserving diffeomorphisms as Poincaré maps for volume preserving flows

Let $q:M\to M$ be a volume-preserving diffeomorphism of a smooth manifold $M$. We study the possibility to present $q$ as the Poincar\'e map, corresponding to a volume-preserving vector field on $\mathbb{T}\times M$, $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.

The trunkenness of a volume-preserving vector field

We construct a new invariant—the trunkenness—for volume-preserving vector fields on up to volume-preserving homeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.

Marbling-based creative modelling

Most mathematical marbling simulations generate patterns for texture mapping and surface decoration. We explore the application of three-dimensional deformations inspired by mathematical marbling as a suite of tools to enable creative shape design. Our tools are expressed as analytical functions of space and are volume-preserving vector fields, meaning that the modelling process preserves volumes and avoids self-intersections. Complicated deformations are easily combined to create complex objects from simple ones. To achieve smooth and high-quality shapes, we also present a mesh refinement and simplification algorithm adapted to our deformations. We show a number of examples of shapes created with our technique in order to demonstrate its power and expressiveness.

Volume preservation by Runge–Kutta methods

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge–Kutta method will respect this property for such systems, but it has been shown by Iserles, Quispel and Tse and independently by Chartier and Murua that no B-Series method can be volume preserving for all volume preserving vector fields. In this paper, we show that despite this result, symplectic Runge–Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge–Kutta methods can preserve a modified measure exactly.

Application of Lie Algebra in Constructing Volume-Preserving Algorithms for Charged Particles Dynamics

AbstractVolume-preserving algorithms (VPAs) for the charged particles dynamics is preferred because of their long-term accuracy and conservativeness for phase space volume. Lie algebra and the Baker-Campbell-Hausdorff (BCH) formula can be used as a fundamental theoretical tool to construct VPAs. Using the Lie algebra structure of vector fields, we split the volume-preserving vector field for charged particle dynamics into three volume-preserving parts (sub-algebras), and find the corresponding Lie subgroups. Proper combinations of these subgroups generate volume preserving, second order approximations of the original solution group, and thus second order VPAs. The developed VPAs also show their significant effectiveness in conserving phase-space volume exactly and bounding energy error over long-term simulations.

A classification of volume preserving generating forms in $\mathbb{R}^3$

In earlier work, Lomeli and Meiss [9] used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. In [20], Xue and Zanna studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators.In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction.We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in $\mathbb{R}^3$.

On volume-preserving vector fields and finite-type invariants of knots

We consider the general non-vanishing, divergence-free vector fields defined on a domain in$3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

Existence of invariant tori in three dimensional maps with degeneracy

We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action–angle–angle maps with degeneracy, satisfying the intersection property. Such degenerate action–angle–angle maps arise upon generic perturbation of three-dimensional volume-preserving vector fields, which are invariant under volume-preserving action of S1 when there is no motion in the group action direction for the unperturbed map. This situation is analogous to degeneracy in Hamiltonian systems. The degenerate nature of the map and the unequal number of action and angle variables make the persistence proof non-standard. The persistence of the invariant tori as predicted by our result has implications for the existence of barriers to transport in three-dimensional incompressible fluid flows. Simulation results indicating existence of two-dimensional tori in a perturbation of swirling Hill’s spherical vortex flow are presented.

The Third Order Helicity of Magnetic Fields via Link Maps

We introduce an alternative approach to the third order helicity of a volume preserving vector field $B$, which leads us to a lower bound for the $L^2$-energy of $B$. The proposed approach exploits correspondence between the Milnor $\bar{\mu}_{123}$-invariant for 3-component links and the homotopy invariants of maps to configuration spaces, and we provide a simple geometric proof of this fact in the case of Borromean links. Based on these connections we develop a formulation for the third order helicity of $B$ on invariant \emph{unlinked} domains of $B$, and provide Arnold's style ergodic interpretation of this invariant as an average asymptotic $\bar{\mu}_{123}$-invariant of orbits of $B$.

Asymptotic Rasmussen invariant

We use simple properties of the Rasmussen invariant of knots to study its asymptotic behaviour on the orbits of a smooth volume preserving vector field on a compact domain of the 3-space. A comparison with the asymptotic signature allows us to prove that asymptotic knots of non-zero invariant are non-alternating. To cite this article: S. Baader, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Volume–preserving Fields and Reeb Fields on 3–manifolds

Volume–preserving fieldX on a 3–manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω. The Reeb vector field of a contact form is of volume–preserving, but not conversely. On the basis of Geiges–Gonzalo’s parallelization results, we obtain a volume–preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume–preserving fields. From many aspects, we discuss the distinction between volume–preserving fields and Reeb–like fields. We establish a duality between volume–preserving fields and h–closed 2–forms to understand such distinction. We also give two kinds of non–Reeb–like but volume–preserving vector fields to display such distinction.

Stochastic continuity equation and related processes

We consider the processes defined by a Langevin equation and the associated continuity equation. The average of the density function, solution of the continuity equation, satisfies the Fokker–Planck equation. For a volume preserving vector field the same equation is satisfied by the average of the integer powers of the density, which are the moments of the related probability density. For a generic vector field the Fokker–Planck equation for the moments is slightly modified. We first illustrate the problem in the simple case of a free particle subject to a white noise, since the averages can be computed by an elementary procedure using the factorization property of the correlation functions of the noise. The probabilistic meaning of the moments is discussed and the comparison between the analytical results and the numerical simulation is shown. The case of a generic Langevin equation is treated by computing the averages via a Dyson expansion after observing that, for a volume preserving vector field, any power of the density function satisfies the same continuity equation with the appropriate initial conditions. As a consequence the results obtained for the free particle are easily extended. An alternative approach is based on the characteristics of the continuity equation; the probability density of this process in an extended phase space still satisfies a Fokker–Planck equation and its moments coincide with the previous definition.

Plane fields related to vector fields on 3-manifolds

This paper is a small collection of results about the topology of non-singular plane fields which are either transverse or tangent to nonsingular volume preserving vector fields on 3-manifolds. Emphasis is given to contact plane distributions and to restrictions of Hamiltonian vector fields to hypersurfaces in symplectic 4-manifolds.

Coset construction of Spin(7), G2 gravitational instantons

We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy Spin(7), G2. We concentrate on the metrics which are defined on R × G/H. If the homogeneous coset spaces G/H have weak G2, SU(3) holonomy, the manifold R × G/H may have Spin(7), G2 holonomy metrics. Using the formulation with vector fields, we investigate the metrics with Spin(7) holonomy on R × Sp(2)/Sp(1), R × SU(3)/U(1). We have found the explicit volume-preserving vector fields on these manifolds using the elementary coordinate parametrization. This construction is essentially dual for solving the generalized self-duality condition for spin connections. We present the most general differential equations for each coset. Then, we develop a similar formulation in order to calculate metrics with G2 holonomy.

Spin(7) holonomy manifold and superconnection

We discuss the higher-dimensional generalization of gravitationalinstantons by using volume-preserving vector fields. We give specialattention to the case of eight dimensions and present a new construction ofthe Ricci-flat metric with holonomy in Spin(7). An example of the metricis explicitly given. Furthermore, it is shown that our formulation has a naturalinterpretation in the Chern-Simons theory written by the language ofsuperconnections.

Signature asymptotique d'un champ de vecteurs en dimension 3

Consider a volume preserving vector field defined in some compact domain of 3-space and tangent to its boundary. A long piece of orbit can be made into a knot by connecting its endpoints by some arc whose length is less than the diameter of the domain. In this paper, we study the behaviour of the signatures of these knots as the lengths of the pieces of orbits go to infinity. We relate this "asymptotic signature" to the "asymptotic Hopf invariant" that has been studied by Arnold.

Smooth normalization of a volume preserving vector field via a canonical transformation

Smooth normalization of a volume preserving vector field via a canonical transformation

Applications of the Ashtekar gravity to four-dimensional hyperkähler geometry and Yang–Mills instantons

The Ashtekar–Jacobson–Smolin–Mason–Newman equations are used to construct the hyperkähler metrics on four-dimensional manifolds. These equations are closely related to anti-self-dual Yang–Mills equations of the infinite-dimensional gauge Lie algebras of all volume-preserving vector fields. Several examples of hyperkähler metrics are presented through the reductions of anti-self-dual connections. For any gauge group anti-self-dual connections on hyperkähler manifolds are constructed using the solutions of both Nahm and Laplace equations.