Harmonic polynomial morphisms between Euclidean spaces

Two non-existence theorems on harmonic polynomial morphisms between Euclidean spaces have been shown.     

A Theorem of Liouville Type for Harmonic Morphisms

Let : M N be a harmonic morphism from a complete noncompact Riemannian manifold M with nonnegative Ricci curvature to a complete Riemannian manifold N with nonpositive scalar curvature. We show if the energy of is finite, then is constant. This can be compared with a similar result for harmonic maps when N has nonpositive sectional curvature due to Schoen and Yau.     

Harmonic morphisms with one-dimensional fibres on Einstein manifolds

We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.

     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

It is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too.     

A perturbation result for q-open quotient morphisms in normed spaces and applications to linear relations

Using some techniques of perturbation theory for quotient morphisms between Banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q-open quotient morphism under small (with respect to the gap topology) perturbations with quotient morphisms. As an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Harmonic Analysis and Degenerate Diffusions on Euclidean Groups

We take up an idea, introduced by Chirikjian andKyatkin, of analyzing a family of left-invariant diffusion equations on Euclidean groups via the group Fourier transform. These diffusion equations model the probability distribution of the orientation in space of certain polymers in solution, including DNA. We study the evolution equations satisfied by the Fourier coefficients of such a solution. Our main task here is to estimate these Fourier coefficients sufficiently well that one canestimate the error in truncating the group inverse Fourier transform to afinite region.     

A Nonexistence Theorem of Proper Harmonic Morphisms Between Hyperbolic Spaces

We provide a nonexistence theorem of harmonic morphisms between hyperbolic spaces which are proper and C ²-smooth to the boundary.     

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.     

A relation between the LG polynomial and the Kauffman polynomial

We give an explicit formula for the fact given by Links and Gould that a one variable reduction of the LG polynomial coincides with a one variable reduction of the Kauffman polynomial. This implies that the crossing number of an adequate link may be obtained from the LG polynomial by using a result of Thistlethwaite. We also give some evaluations of the LG polynomial.     

Harmonic Maps Over Rings

For the torsion-free modules over noncommutative principal ideal domains von Staudt's theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruple is calculated.     

Harmonic Diffeomorphisms between Complete Surfaces

We prove an existing theorem of homotopic harmonic diffeomorphisms between complete surfaces of finite total curvature and nontrivial genus. This is a generalization of a theorem of Jost and Schoen [4].     

P-调和映射流的存在性与不唯一性

构造了ｐ 调和发展方程的初边值问题的一个整体弱解；证明了当初值ｕ０（ｘ）是弱ｐ 调和映射但不是弱ｐ 驻调和映射时问题弱解的不唯一性．     

稳定调和映照的不存在性

讨论了从Finsler流形到Riemannian流形的稳定调和映照的不存在性,得到了一个拼挤定理.     

A Bridge Principle for Harmonic Maps

We prove a bridge principle for harmonic maps between generalmanifolds.     

Harmonic and monogenic potentials in low dimensional Euclidean half‐space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary were also determined. In this paper, the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down. Copyright © 2013 John Wiley & Sons, Ltd.     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ¹ into ΩG are related to Yang-Mills G-fields on ℝ⁴. This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics”     

Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。