On non-isotropic harmonic maps of surfaces into complex projective spaces

It is proved by purely algebraic method that weakly conformai, conformai andA _z ³ = 0 are mutually equivalent if ϕ :Ω→ℂP ⁿ is a non-isotropic harmonic map and the harmonic maps with isotropy order ≥3 are uniquely determined by a system of ordinary differential equations. A method is given, by which the isotropy orders of non-isotropic harmonic maps can be computed.     

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.     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

It is proved that any harmonic map ϕ : Ω →Sp(N) from a simply connected domain Ω ⊆R ²⋃ | ∞ | into the symplectic groupSp(N) ⊂U(2N) with finite uniton number can be factorized into a product of a finite number of symplectic unitons. Based on this factorization, it is proved that the minimal symplectic uniton number of ϕ is not larger thanN, and the minimal uniton number of ϕ is not larger than 2N - 1. The latter has been shown in literature in a quite different way.     

On harmonic maps between generalized flag manifolds

The theory of harmonic maps has been developed since the 1960's (see [2]). In recent years, some authors discussed the harmonicity of “homogeneous” maps between Riemannian homogeneous spaces using the theory of Lie groups. LetG andG′ be compact Lie groups,H andH′ their closed subgroups respectively. Assume that a homomorphism θ:G→G′ mapsH intoH′; then there exists an induced mapf _θ:G/H→G′/H′. M.A. Guest gave a necessary and sufficient condition for such a map to be harmonic, whenG/H andG′/H′ are generalized flag manifolds,H=T is a maximal torus andG′ is a unitary group; and he gave some interesting examples (see [3]). We generalize his results to the case of general generalized flag manifoldsG/H, i.e.H is a centralizer of a torus, and give some new examples of harmonic maps. Supported in part by the National Natural Science Foundation of China and K.C. Wong Education Foundation (in Hong Kong).     

Topological restrictions for circle actions and harmonic morphisms

 Let M ^m be a compact oriented smooth manifold admitting a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M ^m are zero and its Euler number is nonnegative and even. In particular, M ^m has signature zero. We apply this to obtain non-existence of harmonic morphisms with one-dimensional fibres from various domains, and a classification of harmonic morphisms from certain 4-manifolds. Received: 16 May 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 58E20, 53C43, 57R20.     

A finiteness theorem of harmonic maps from compact lie groups toO HS(H)

In this article we study the behavior of harmonic maps from compact connected Lie groups with bi-invariant metrics into a Hilbert orthogonal group. In particular, we will demonstrate that any such harmonic map always has image contained within someO(n),n<∞. Since homomorphisms are a special subset of the harmonic maps we get as a corollary an extension of the Peter-Weyl theorem, namely, that every representation of a connected compact Lie group is finite dimensional. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

2D semiclassical model for high harmonic generation from gas

The electron behavior in laser field is described in detail. Based on the ID semiclassical model, a2D semiclassical model is proposed analytically using 3D DC-tunneling ionization theory. Lots of harmonic features are explained by this model, including the analytical demonstration of the maximum electron energy 3.17U _p Finally, some experimental phenomena such as the increase of the cutoff harmonic energy with the decrease of pulse duration and the “anomalous” fluctuations in the cutoff region are explained by this model.     

Jacobi Structures on Affine Bundles

Abstract We study affine Jacobi structures （brackets） on an affine bundle π ： A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^＋ = ∪p∈M Aff（Ap, R） of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^＋. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited （strongly-affine or affine-homogeneous Jacobi structures） over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.     

Chern-simons theory in the SO(5)/U(2) harmonic superspace

We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposing these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to the D=3, N=6 conformal supersymmetry realized on N=5 superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 217–234, November, 2008.     

Transversal harmonic transformations for Riemannian foliations

The main purpose of the present paper is to study geometric properties of transversal (infinitesimal) harmonic transformations for Riemannian foliations. For the point foliation these notions are discussed in [14]. Especially we treat transversal infinitesimal harmonic transformations from the standpoint of λ-automorphisms. Our results extend those obtained in [6, 7, 15] for the case of harmonic foliations. Mathematics Subject Classifications (2000): Primary 53C20, Secondary 57R30.     

Sequences of harmonic maps in the 3‐sphere

We define two transforms of non‐conformal harmonic maps from a surface into the 3‐sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3‐sphere, H‐surfaces in Euclidean 3‐space and almost complex surfaces in the nearly Kähler manifold . As a consequence we can construct sequences of H‐surfaces and almost complex surfaces.     

Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤ_q≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph . More generally, (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees. Mathematics Subject Classifications (2000) 60J50, 05C25, 20E22, 31C05, 60G50. Supported by European Commission, Marie Curie Fellowship HPMF-CT-2002-02137.     

Approximation by harmonic functions in theC m -Norm and harmonicC m -capacity of compact sets in ℝ n

We study the function Λ ^m (X), 0<m<1, of compact setsX in ℝ ⁿ , n≥2, defined as the distance in the spaceC ^m (X)≡lip ^m(X) from the function |x|² to the subspaceH ^m (X) which is the closure inC ^m (X) of the class of functions harmonic in the neighborhood ofX (each function in its own neighborhood). We prove the equivalence of the conditions Λ ^m (X)=0 andC ^m (X)=H ^m (X). We derive an estimate from above that depends only on the geometrical properties of the setX (on its volume). Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 372–382, September, 1997. Translated by I. P. Zvyagin     

Hyperbolic <Emphasis Type="Italic">n</Emphasis>-dimensional manifolds with automorphism group of dimension <Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>

We obtain a complete classification of complex Kobayashihyperbolic manifolds of dimension n ≥ 2, for which the dimension of the group of holomorphic automorphisms is equal to n². Received: May 2005 Accepted: November 2005     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Symmetric Harmonic Sheaves Possessing Bipotentials

Let be the family of sheaves H of continuous functions on a Brelot harmonic space with a countable base such that locally the Dirichlet problem with respect to H is solvable, H satisfies Harnack inequalities and also H has a symmetry property. Defining the notions of H-biharmonic functions, H-biharmonic Green functions, H-bipotentials, H-biharmonic extensions, etc. we study the interrelation between them and exhibit various classifications of the family of sheaves.     

The number of components of complements to level surfaces of partially harmonic polynomials

In this paperk-harmonic polynomials in ℝ ⁿ , i.e. polynomials satisfying the Laplace equation with respect tok variables: ∂²/∂x ₁ ² +...+∂²/∂x _k ² F=0 are considered; here 1≤k≤n andn≥2. For a polynomialF (of degreem) of this type, it is proved that the number of components of the complements of its level sets does not exceed 2m ⁿ⁻¹+O(m ⁿ⁻²). Under the assumptions that the singular set of the level surface is compact or that the leading homogeneous part of thek-harmonic polynomialF is nondegenerate, sharper estimates are also established. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 831–835, December, 1997 Translated by S. S. Anisov     

Invariant percolation and harmonic Dirichlet functions

The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the random-cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for random-cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of ℓ² Betti numbers. Received: May 2004 Revised: March 2005 Accepted: May 2005