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.

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].

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.

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.

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.

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.

Parameterization Transfer for a Planar Computational Domain in Isogeometric Analysis

In this paper,we propose a parameterization transfer algorithm for planar domains bounded by B-spline curves,where the shapes of the planar domains are similar.The domain geometries are considered to be similar if their simplified skeletons have the same structures.One domain we call source domain,and it is parameterized using multi-patch B-spline surfaces.The resulting parameterization is C1 continuous in the regular region and G1 continuous around singular points regardless of whether the parameterization of the source domain is C1/G1 continuous or not.In this algorithm,boundary control points of the source domain are extracted from its parameterization as sequential points,and we establish a correspondence between sequential boundary control points of the source domain and the target boundary through discrete sampling and fitting.Transfer of the parametrization satisfies C1/G1 continuity under discrete harmonic mapping with continuous constraints.The new algorithm has a lower calculation cost than a decomposition-based parameterization with a high-quality parameterization result.We demonstrate that the result of the parameterization transfer in this paper can be applied in isogeometric analysis.Moreover,because of the consistency of the parameterization for the two models,this method can be applied in many other geometry processing algorithms,such as morphing and deformation.

曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

On Harmonic K-Quasiconformal Mappings Associated with Asymmetric Vertical Strips

In this paper,we discuss the sense-preserving univalent harmonic mappings from the unit disk D onto asymmetrical vertical strips Ωα={ω：α-π/2sinαR（ω）α/2sinα},π/2≤απSuch results as analytic representation formula,coefficient estimates,distortion theorem and area theorem are derived.

SURFACE MESH PARAMETERIZATION WITH NATURAL BOUNDARY

Using the projected curve of surface mesh boundary as parameter domainborder, linear mapping parameterization with natural boundary is realized. A fast algorithm forleast squares fitting plane of vertices in the mesh boundary is proposed. After the mesh boundary isprojected onto the fitting plane, low-pass filtering is adopted to eliminate crossovers, sharpcorners and cavities in the projected curve and convert it into an eligible convex parameter domainboundary. In order to facilitate quantitative evaluations of parameterization schemes, threedistortion-measuring formulae are presented.

On the Application of a Mobile Grid Technology to Computational Fluid Dynamics

On the basis of harmonic mapping theory,a mobile grid technology is applied to computational fluid dynamics(CFD).Starting from the observation that standard fixed-grid techniques often fail in addressing problems with large deformations,we elaborate a new algorithm relying on the software COMSOL Multiphysics 5.3a to solve the coupling of the mobile grid equation and the governing differential equations for fluid flow.The motion of water in a water tank when the tank waggles is simulated.We demonstrate that this technology can be implemented without a significant increase in the computational cost with respect to standard numerical methods.

TANGENT UNIT-VECTOR FIELDS:NONABELIAN HOMOTOPY INVARIANTS AND THE DIRICHLET ENERGY

Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we compute the infimum Dirichlet energy 6（H） for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for C（H） involves a topological invariant - the spelling length - associated with the （non-abelian） fundamental group of the n-times punctured two-sphere, π1（S2 - {s1,..., sn}, ＊）. The lower bound for C（H） is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes （classes containing wholly conformal and anticonformal representatives respectively）, the expression for C（H） reduces to a previous result involving the degrees of a set of regular values sl,…… sn in the target 82 space. These degrees may be viewed as invariants associated with the abelianization of vr1（S2 - {s1,..., sn}, ＊）. For nonconformal classes, however, ε（H） may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy （one-constant Oseen-Frank energy） of reflection-symmetric tangent unitvector fields in a rectangular prism.

Koebe Type Theorems and Pre-Schwarzian of Harmonic K-quasiconformal Mappings,and Their Applications

In this article,we first establish an asymptotically sharp Koebe type covering theorem for harmonic K-quasiconformal mappings.Then we use it to obtain an asymptotically Koebe type distortion theorem,a coefficients estimate,a Lipschitz characteristic and a linear measure distortion theorem of harmonic K-quasiconformal mappings.At last,we give some characterizations of the radial John disks with the help of pre-Schwarzian of harmonic mappings.

Some Properties of the Beurling-Ahlfors Extension

In this paper, we study the Beurling-Ahlfors extensions and prove two results. The first variation of the Beurling-Ahlfors extension is not always harmonic; the Beurling-Ahlfors extension of a quasisymmetric mapping is not always harmonic.

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.

Mapping Problems,Fundamental Groups and Defect Measures

We study all the possible weak limits of a minimizing sequence,for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition.We show that if p is not an integer,then any such weak limit is a strong limit and,in particular,a stationary p-harmonic map which is C1,α continuous away from a closed subset of the Hausdorff dimension ≤n-[p]-1.If p is an integer,then any such weak limit is a weakly p-harmonic map along with a（n-p）-rectifiable Radon measure μ.Moreover,the limiting map is C1,α continuous away from a closed subset ∑=spt μ∪S with Hn-p（S）=0.Finally,we discuss the possible varifolds type theory for Sobolev mappings.