ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR

In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions.Then,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F),where F has bounded length distortion and L is a differential operator.

THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL

In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.

Some Theorems About Unitons of Harmonic Maps into U(N)

By using the simplified method of factorization given by Valli, and the correspondence between the harmonic map φ∶S 2→U(N) and U(N) uniton bundle ν(φ) with energy corresponding to the bundles’ second Chern class, which is established by Anand, the energy in a case φ∶S 2→U(N) is investigated in order to estimate the energy of a uniton using the uniton number. It is proved that Uhlenbeck’s factorization is energy decreasing. And a method of estimating the energy of a uniton by the uniton number is given.

Heat flow of harmonic maps from noncompact manifolds

The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.

Some results on P-harmonic maps and exponentially harmonic maps between Finsler manifolds

This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].

VANISHING THEOREMS FOR ACH KHLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincare inequality case and establish a vanishing theorem provided that the weighted function p is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincare inequality.

HARMONIC MAPPINGS OF THE HEXAGASKET TO THE CIRCLE

Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].

The Energy Growth Property for Harmonic Maps

In this paper,we study the energy growth property for a class of harmonic maps. A special estimation of energy growth is obtained.

Some Results of Biharmonic Maps

In this paper, we investigate biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain some non-existence results for these maps.

A REMARK ON THE QUASI-HARMONIC SPHERES

Several theorems on the finiteness of energy for quasi- harmonic spheres are proved, some counter- examples which state thatthe energy of quasi- harmonic sphere may be infinite are given.The results support some conditions of a question posed by Lin Fanghua and Wang Changyou.

Symmetrically Harmonic Kaluza-Klein Metrics on Tangent Bundles

Let (M, g) be a Riemannian manifold and G be a Kaluza-Klein metric on its tangent bundle TM. A metric H on TM is said to be symmetrically harmonic to G if the metrics G and H are harmonic w.r.t. each other;that is the identity maps id: (TM,G) → (TM,H) and id: (TM,H) → (TM,G) are both harmonic maps. In this work we study Kaluza-Klein metrics H on TM which are symmetrically harmonic to G. In particular, we characterize and determine horizontally and vertically conformal Kaluza-Klein metrics H on TM, which are symmetrically harmonic to G.

Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough.

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

In this paper, we explore the linear combinations of right half-plane mappings and vertical strip mappings. We demonstrate that the combinations of these harmonic mappings are convex in the vertical direction provided they are locally univalent and sense-preserving. Furthermore, we extend this analysis to a more general case by setting specific conditions. Additionally, we take some common parameters such as as the dilatation of these harmonic mappings, and prove the sufficient conditions that their combinations are locally univalent and convex in the vertical direction. Several examples are constructed by the Mathematica software to demonstrate our main results.

Estimates on Bloch constants for planar harmonic mappings

The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Grigoryan.

On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals

For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

The Schwarz-Pick lemma for planar harmonic mappings

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)).

ON HARMONIC MAPS INTO SYMPLECTIC GROUPS Sp(N)

By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic ex--tended uniton are introduced. The method of the symplectic Backlund transformation and the Darboux transformation is used to construct new symplectic unitons from a known one.

The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = （-1, 1）, AU（z）/cosU（z）π/2≤4/π 1/1-｜z｜^2 holds for z E D, where Au（z） is the maximum dilation of U at z. The inequality is sharp for any z E D and any value of U（z）, and the equality occurs for some point in D if and only if U（z） = 4Re {arctan ~a（z）}, z E D, with a M＆bius transformation φa of D onto itself.

Left Invariant Harmonic Vector Fields on Solvable Lie Groups

It's shown that on an n-dimensional LCS Lie group,there exist harmonic sections and left invariant vector fields defining harmonic maps only when it is endowed with a flat left invariant pseudo-Riemannian metric.Moreover,we determine all the vector fields which are critical points of the energy functional restricted to vector fields of the same length on the n-dimensional pseudo-Riemannian LCS Lie group.