The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps ...The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps into convex balls.展开更多
Darboux transformation method is used for constructing harmonic maps from R2 to U(N).The explicit expressions for Darboux matrices are used to obtain new harmonic maps from aknown one.The algorithm is purely algebraic...Darboux transformation method is used for constructing harmonic maps from R2 to U(N).The explicit expressions for Darboux matrices are used to obtain new harmonic maps from aknown one.The algorithm is purely algebraic and can be repeated successively to obtain aninfinite sequence of harmonic maps. Single and multiple solitons are obtained with geometriccharacterizations and it is proved that the interaction between solitons is elastic. By introducingthe singlllar Darboux transformations, an explicit method to construct new unitons is presented.展开更多
The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex do...The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).展开更多
文摘The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps into convex balls.
文摘Darboux transformation method is used for constructing harmonic maps from R2 to U(N).The explicit expressions for Darboux matrices are used to obtain new harmonic maps from aknown one.The algorithm is purely algebraic and can be repeated successively to obtain aninfinite sequence of harmonic maps. Single and multiple solitons are obtained with geometriccharacterizations and it is proved that the interaction between solitons is elastic. By introducingthe singlllar Darboux transformations, an explicit method to construct new unitons is presented.
基金supported by National Natural Science Foundation of China (Grant No.10671093)
文摘The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).
