Functions which are smooth along vector fields

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 95–102, July, 1990.     

Constructions of harmonic polynomial maps between spheres

The complexity of _q -eigenmaps, i.e. homogeneous degreeq harmonic polynomial mapsf:S ^m S ⁿ, increases fast with the degreeq and the source dimensionm. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of ₂-eigenmaps.     

Real, complex and quaternionic equivariant vector fields on spheres

The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J-groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Önder [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.     

Positive vector bundles and harmonic maps

Sunto Si provano alcuni risultati sul carattere analitico complesso di applicazioni armoniche fra varietà Kähleriane, utilizzando due differenti tecniche entrambe basate sulla teoria dei fibrati positivi: nel caso compatto, si usa uno sviluppo generalizato alla Bochner- Kodaira del Laplaciano complesso , utilizzando il fatto che, se f: N M è armonica, allora f è una (0, 1) -forma - armonica a valori nel fibrato indotto E=f^*[TM]_1,0; nel caso aperto, se f è stabile, il problema viene ricondotto allo studio dell'equazione con stime L² pesate: sotto opportune ipotesi, il fibrato E può essere dotato di una struttura di fibrato olomorfo positivo che trasforma in .     

Characterizing spheres by conformal vector fields

In this short note we consider an n-dimensional compact Riemannian manifold (M, g) of constant scalar curvature S = n(n − 1)c and show that the presence of a nontrivial conformal vector field ξ on M forces S to be positive. Then we show that an appropriate control on the energy of ξ makes M to be isometric to the n-sphere S ⁿ (c).     

Combinatorial construction of tangent vector fields on spheres

For every odd n, on the sphere S ⁿ , ρ(n) ? 1 linear orthonormal tangent vector fields, where ρ(n) is the Hurwitz-Radon number, are explicitly constructed. For each 8 × 8 sign matrix, compositions for infinite-dimensional positive definite quadratic forms are explicitly constructed. The infinite-dimensional real normed algebras thus arising are proved to have certain properties of associativity and divisibility type.     

Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

稳态轴对称引力场与调和映射

本文证明了广义相对论中每个稳态轴对称真空场都给出从R³到H²的调和映射,反之也可从调和映射作出爱因斯坦方程的稳态轴对称真空解。     

On annihilators of harmonic vector fields

Let Ω⊂ℝ^N be a smooth bounded domain. We characterize smooth vector fields g on ∂Ω which annihilate all harmonic vector fields f in Ω continuous up to ∂Ω, with respect to the pairing (dσ denotes the hypersurface measure on ∂Ω). In addition, we extend these results to differential forms with harmonic vector fields being replaced by harmonic fields, i.e., forms f satisfying df=0, δf=0. A smooth form g on ∂Ω is an annihilator if and only if suitable extensions, u and v, into Ω of its normal and tangential components on ∂Ω, satisfy the generalized Cauchy-Riemann system du=δv, δu=0, dv=0 in Ω. Finally, we prove that the described smooth annihilators are weak^* dense among all annihilators. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 90–108.     

Instability of Hopf vector fields on Lorentzian Berger spheres

In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.     

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let X ì \mathbb Rⁿ{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in \mathbbC \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 ⊂ \mathbbR³ \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 ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C ¹ with [`(?)] f\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.