Harmonic Vector Fields Research Articles

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M m of a Euclidean space R m+p is said to have harmonic mean curvature vector field if $${\Delta \vec{H}=0}$$ , where $${\vec{H}}$$ is the mean curvature vector field of $${M\hookrightarrow R^{m+p}}$$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.

Klein bottle logophysics: a unified principle for non-linear systems, cosmology, geophysics, biology, biomechanics and perception

We present a unified principle for science that surmounts dualism, in terms of torsion fields and the non-orientable surfaces, notably the Klein Bottle and its logic, the Möbius strip and the projective plane. We apply it to the complex numbers and cosmology, to non-linear systems integrating the issue of hyperbolic divergences with the change of orientability, to the biomechanics of vision and the mammal heart, to the morphogenesis of crustal shapes on Earth in connection to the wavefronts of gravitation, elasticity and electromagnetism, to pattern recognition of artificial images and visual recognition, to neurology and the topographic maps of the sensorium, to perception, in particular of music. We develop it in terms of the fundamental 2:1 resonance inherent to the Möbius strip and the Klein Bottle, the minimal surfaces representation of the wavefronts, and the non-dual Klein Bottle logic inherent to pattern recognition, to the harmonic functions and vector fields that lay at the basis of geophysics and physics at large. We discuss the relation between the topographic maps of the sensorium, and the issue of turning inside-out of the visual world as a general principle for cognition, topological chemistry, cell biology and biological morphogenesis in particular in embryology

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Total-angular-momentum (TAM) waves provide a set of basis functions for scalar, vector, and tensor fields that can be used in place of plane waves and that reflect the rotational symmetry of the spherical sky. Here we discuss three-point correlation functions, or bispectra in harmonic space, for scalar, vector, and tensor fields in terms of TAM waves. The Wigner-Eckart theorem dictates that the expectation value, assuming statistical isotropy, of the product of three TAM waves is the product of a Clebsch-Gordan coefficient (or Wigner-3j symbol) times a function only of the total-angular-momentum quantum numbers. Here we show how this works, and we provide explicit expressions relating the bispectra for TAM waves in terms of the more commonly used Fourier-space bispectra. This formalism will be useful to simplify calculations of projections of three-dimensional bispectra onto the spherical sky.

Uniqueness and non-uniqueness in the non-axisymmetric direction problem

We consider the following nonlinear boundary-value problem in the exterior space b V = {x 2 R 3 : |x| > 1} of the unit sphere S: given a vector field D : S ! R 3 , we ask for all harmonic vector fields B : b V ! R 3 vanishing at infinity and parallel to D on S, i.e. there is an amplitude a : S ! R such that B = aD. This ‘geomagnetic direction problem’ is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth’s surface. Here we investigate this problem using the well-known multipole expansion B = P1 n=1 Pn k= n c k B k of exterior harmonic vector fields. In the case that only finite linear combinations of the B k are admissible solutions we prove uniqueness of the problem. Without this restriction, non-uniqueness of the problem is well established in the axisymmetric case. In the non-axisymmetric case, we prove here uniqueness for all direction fields D k := B k |S (n 2 N, 0 < |k| n), but give also an example for non-uniqueness.

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ξ, when ∇ξ is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ξ is geodesic, ξ is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures σ are harmonic sections, in the sense of Vergara-Diaz and Wood [25], and in some cases they are also harmonic maps.

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in $$ \mathbb{C}$$ are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ $$ \mathbb{R}^3$$ is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ $$ \mathbb{C}$$ (when f is assumed to be in C 1 with $$\bar \partial {\kern 1pt}f$$ small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.

Flow around an ellipsoid of revolution in a harmonic coaxial vector field

We give an integral equation defining a coaxial magnetic field near the surface of a superconductive axisymmetric body and the velocity of the liquid near the surface of an axisymmetric body situated coaxially to the flow of an ideal liquid. Using this equation in the case when the axisymmetric magnetic field before the placement of an ellipsoid of revolution coaxially to the field changed along the axis by a polynomial law, we analytically define the densities of the surface current and the force with which the magnetic field acts on the ellipsoid. Also the velocity of the liquid is determined near the surface of the ellipsoid of revolution and the force acting on the ellipsoid placed coaxially in the flow of an ideal liquid when the velocity of the liquid before the placement of the ellipsoid changed along the axis of symmetry by a polynomial law.

Ellipsoid flowed around by a harmonic vector field

We consider the screening of an external magnetic field in which a superconducting ellipsoid is inserted and a change in the velocity distribution in an ideal liquid flowing around an ellipsoid inserted in it. In both cases, the solution is given by a harmonic vector field parallel to the surface near the ellipsoid.

Characterization of Interictal Epileptiform Discharges with Time-Resolved Cortical Current Maps Using the Helmholtz–Hodge Decomposition

Source estimates performed using a single equivalent current dipole (ECD) model for interictal epileptiform discharges (IEDs) which appear unifocal have proven highly accurate in neocortical epilepsies, falling within millimeters of that demonstrated by electrocorticography. Despite this success, the single ECD solution is limited, best describing sources which are temporally stable. Adapted from the field of optics, optical flow analysis of distributed source models of MEG or EEG data has been proposed as a means to estimate the current motion field of cortical activity, or “cortical flow.” The motion field so defined can be used to identify dynamic features of interest such as patterns of directional flow, current sources, and sinks. The Helmholtz–Hodge Decomposition (HHD) is a technique frequently applied in fluid dynamics to separate a flow pattern into three components: (1) a non-rotational scalar potential U describing sinks and sources, (2) a non-diverging scalar potential A accounting for vortices, and (3) an harmonic vector field H. As IEDs seem likely to represent periods of highly correlated directional flow of cortical currents, the U component of the HHD suggests itself as a way to characterize spikes in terms of current sources and sinks. In a series of patients with refractory epilepsy who were studied with magnetoencephalography as part of their evaluation for possible resective surgery, spike localization with ECD was compared to HHD applied to an optical flow analysis of the same spike. Reasonable anatomic correlation between the two techniques was seen in the majority of patients, suggesting that this method may offer an additional means of characterization of epileptic discharges.

Harmonic vector fields on compact Lorentz surfaces

We study harmonic vector fields on a Lorentzian torus T2 i.e. critical points of the total bending functional \({\mathcal {B} : \mathcal {E} \to \mathbb {R}}\) were \({\mathcal {E}}\) consists of all unit timelike vector fields on T2. We derive the first variation formula for \({\mathcal {B}}\) in terms of the Lorentz angle function associated to each \({X \in \mathcal {E}}\) and give applications on flat Lorentzian tori.

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.

Instability of the geodesic flow for the energy functional

Let (S n (r),g 0 ) be the canonical sphere of radius r. Denote by G s the Sasaki metric on the unit tangent bundle T 1 S n (r) induced from g 0 and by G s the Sasaki metric on T 1 T 1 S n (r) induced from G s . We resolve here, for n ≥ 7, a question raised by Boeckx, Gonzalez-Davila, and Vanhecke: namely, we prove that the geodesic flow ξ: (T 1 S n (r), G s ) → (T 1 T 1 S n (r), G s ) is an unstable harmonic vector field for any r > 0 and n ≥ 7. In particular, in the case r = 1, ξ is an unstable harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric G s .

Legendre Surfaces with Harmonic Mean Curvature Vector Field in the Unit $5$-Sphere

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Pseudoharmonic maps and vector fields on CR manifolds

Building on the work by J. Jost and C.-J. Xu (32), and E. Barletta et al. (3), we study smooth pseudoharmonic maps from a compact strictly pseudoconvex CR manifold and their generalizations e.g. pseudoharmonic unit tangent vector fields. The purpose of this paper is to study several analogs to differential geometric objects appearing in Riemannian geometry and admitting a treatment based on elliptic theory e.g. the Laplace-Beltrami operator (cf. (40)), harmonic maps among Riemannian manifolds (cf. (49)), and harmonic vector fields (regarded as smooth maps of a Riemannian manifold into the total space of the tangent bundle endowed with the Sasaki metric, cf. (51 )a nd (52)). We obtain the following results. Boundary values of Bergman-harmonic maps � :� ! S from a smoothly bounded strictly pseudoconvex domain � � C n into a Riemannian manifoldS are shown to be pseudoharmonic maps, provided their normal derivatives vanish. We prove that @b-pluriharmonic maps are pseudoharmonic maps. A pseudoharmonic map� : M ! Sfrom a compact strictly pseudoconvex CR manifold into a sphere is shown either to link or to meet any codimension 2 totally geodesic sphere in S � . Also we prove that a smooth vector field X : M ! T ðMÞ from a strictly pseudoconvex CR manifold M is a pseudoharmonic map if and only if X is parallel (with respect to the Tanaka-Webster connection) along the maximally complex, or Levi, distribution. We start a theory of pseudoharmonic vector fields i.e. unit vector fields X 2 U ðM;� Þ which are critical points of the energy functional EðX Þ¼ 1 R M traceG� ð� HXS� Þ� ^ð d� Þ n relative to variations through unit vector fields. Any such critical point X is shown to satisfy the nonlinear subelliptic systembX þk r H Xk 2 X ¼ 0 .A lsoinfX2U ðM;� ÞEðX Þ¼ n VolðM;� Þ yet E is unbounded from above. We establish first and second variation formulae for E : U ðM;� Þ!½ 0; þ1Þ and give applications.

Applications of 2-Riemannian Helmholtz-Hodge Decomposition to MEG Source Dynamics

Event Abstract Back to Event Applications of 2-Riemannian Helmholtz-Hodge Decomposition to MEG Source Dynamics Sheraz Khan1*, Julien Lefevre2, Manoj Raghavan1 and Sylvain Baillet1 1 Medical College of Wisconsin, Athena project team, United States 2 University of Mediterranee, LSIS - UMR CNRS 6168 , France MEG sources computed using distributed models exhibit complex spatial and temporal dynamics. We have recently proposed that optical flow techniques could be adapted to the evaluation of basic kinematics of MEG source current on the cortical surface [1, 2]. Helmholtz-Hodge Decomposition (HHD) is a technique that decomposes a 2D (resp. 3D) continuous vector field into a sum of three parts: a non-rotational component deriving from the gradient of a scalar potential U, a non-diverging component deriving from the rotation of a scalar potential A (resp. vectorial potential), and a harmonic vector field whose Laplacian vanishes [3, 4]. Even when described on polyhedral surfaces [5] computations are typically performed locally in Euclidean space. Surface curvature has to be taken into account for a proper estimation of vector fields on tangent spaces [1], and convergence is altered by non-flatness properties. We redefined HHD in Riemannian space which enables us to detect features in motion-fields even on highly-curved surfaces, like the cortex.We demonstrate automatic extraction of salient dynamical features from MEG source activity during a somatosensory stimulation task, thereby facilitating reproducible automated analysis of experimental data. We further show how HHD can be used to characterize the dynamics of focal epileptic activity by finding critical points in the surface distribution of scalar potential U. Divergence provides a quantitative measure of cortical dynamics that appears to reveal areas of initiation of spike and seizure activity within epileptic networks.This methodology and early applications suggest that HHD-based measures can capture complex dynamical features from the time-evolving vector motion field of cortical current flow.

On p-harmonic mappings in the plane

To every p -harmonic vector field in the plane with p ≥ 2 there corresponds a quasilinear system of first order PDEs which couples the complex gradients of the coordinate functions of the field. The ellipticity of such system is proved. A relation between planar quasiregular mappings and p -harmonic fields is discussed. The p -harmonic conjugate problem is stated.

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S 2n+1 is the Reeb vector field of a natural Sasakian structure on S 2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.

Bending and stretching unit vector fields in Euclidean and hyperbolic 3-space

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples considered have infinite total bending.

An Expression of Harmonic Vector Fields on Hyperbolic 3-Cone-Manifolds in Terms of Hypergeometric Functions

Let V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (∆ + 4)τ = 0 , where ∆ is the Laplacian of V and τ is the dual 1-form of v . We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann’s P -equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann’s P -equation.

On the Dirichlet problem for the harmonic vector fields equation

We show that any harmonic (with respect to the Bergman metric) vector field tangent to the Levi distribution of the foliation by level sets of the defining function φ ( z ) = − K ( z , z ) − 1 / ( n + 1 ) of a strictly pseudoconvex bounded domain Ω ⊂ C n which is smooth up to the boundary must vanish on ∂ Ω . If n ≠ 5 and u T is a harmonic vector field with u ∈ C 2 ( Ω ¯ ) then u | ∂ Ω = 0 .