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.

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].

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.

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.

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].

COMPUTING HARMONIC MAPS AND CONFORMAL MAPS ON POINT CLOUDS

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space,using point cloud data only.Given a surface,or a point cloud approximation,we simply use the standard cubic lattice to approximate itsϵ-neighborhood.Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices.The conformal map,or the surface uniformization,is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature.We propose algorithms and numerical examples for closed surfaces and topological disks.To the best of the authors’knowledge,our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.

On the Regularity for Stationary Harmonic Maps

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.

COMBINATION OF GLOBAL AND LOCAL APPROXIMATION SCHEMES FOR HARMONIC MAPS INTO SPHERES

It is well understood that a good way to discretize a pointwise length constraint in partial differential equations or variational problems is to impose it at the nodes of a triangulation that defines a lowest order finite element space. This article pursues this approach and discusses the iterative solution of the resulting discrete nonlinear system of equations for a simple model problem which defines harmonic maps into spheres. An iterative scheme that is globally convergent and energy decreasing is combined with a locally rapidly convergent approximation scheme. An explicit example proves that the local approach alone may lead to ill-posed problems; numerical experiments show that it may diverge or lead to highly irregular solutions with large energy if the starting value is not chosen carefully. The combination of the global and local method defines a reliable algorithm that performs very efficiently in practice and provides numerical approximations with low energy.

Linear Connectivity, Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz-Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.

Foliations associated to harmonic maps on some complex two ball quotients

This article is an attempt to understand harmonic and holomorphic maps between two bounded symmetric domains in special situations. We study foliations associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex two ball quotients.Some open questions are raised as well.

A global weak solution to the Lorentzian harmonic map flow

We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric.We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method.A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis.Moreover,the moving mesh method has finite time blowup when the underlying continuous problem does.In situations where the continuous problem has infinite time blowup,the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases.The inadequacy of a uniform mesh solution is clearly demonstrated.

Existence for V T-harmonic maps from compact manifolds with boundary

We consider a kind of generalized harmonic maps,namely,the V T-harmonic maps.We prove an existence theorem for the Dirichlet problem of V T-harmonic maps from compact manifolds with boundary.

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 from Kahler Manifolds

Some Liouville type theorems for harmonic maps from Kahler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (except IV（2）) to any Riemannian manifold with finite energy has to be constant.

Continuity of Almost Harmonic Maps with the Perturbation Term in a Critical Space

The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class(Zygmund class)L^(n/2)log^(q) L,n is the dimension with n≥3.They prove that when q>n/2 the solution must be continuous and they can get continuity modulus estimates.As a byproduct of their method,they also study boundary continuity for the almost harmonic maps in high dimension.

Harmonic maps with torsion

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion.Such connections have already been classified in the work of Cartan(1924).The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges.We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.

On the Construction of Some New Harmonic Maps from R^m to H^m

In this note, we const, ruct some new harmonic maps from R^m to H^m via symmetric methods.