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.     

Construction of differentiable transformations

Construction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

Biharmonic Maps from Tori into a 2-Sphere

Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.     

Uniqueness and stability of harmonic maps and their Jacobi fields

Let M, N be Riemannian manifolds and let f₁, f₂: MN be harmonic maps. Using a maximum principle, an estimate of the distances of these maps by the distances of their boundary values will be proved. Corresponding estimates will be stated for the norm of Jacobi fields along harmonic maps, and for the distances of solutions of the heat equation.Dedicated to Hans Lewy and Charles B. Morrey     

The Jacobian, the square root and the set ∞

In these notes, we wish to present a new approach to the problem of prescribing Jacobian determinants in dimension two: we restrict ourselves to the case the datum is a finite sum of Dirac masses. The main point is to show that we may relate this problem to the search of harmonic maps into a singular space shaped as the symbol ∞. The later problem in turn is closely linked to questions in complex analysis. A large part of the paper is devoted to a presentation of these mathematical objects and their connections.     

EXPLICIT CONSTRUCTION OF HARMONIC MAPS FROM R^2 TO U（N）

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Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

带自由边界调和映射的Liouville型定理

本文给出一类带由边界的调和映射的Liouville型定理,这种类型的定理在微分几何的一些问题中有十分重要的应用.我们通过对调和映射的能量选取特殊的变分族,得到任意从半空间的简单流形到一黎曼流形的带自由边界的调和映射在如果满足适当的条件(见定理)必为常值映射的结果.     

$\bar \partial $‐energy integral and uniqueness of harmonic maps

In this paper, we introduce a new energy functional, ‐energy functional, instead of the total energy functional to investigate the uniqueness of harmonic maps with respect to any given metric on the unit disk. Even in the setting that the Hopf differentials of harmonic maps are not integrable, certain uniqueness theorems of harmonic maps are obtained, which improve a result due to Markovi? and Mateljevi? in 1999. Moreover, a generalized energy‐minimizing property of harmonic maps is discussed. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

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.     

Harmonic Maps and Von Staudt’s Theorem Over Rings

We use U. Brehm’s ideas to study cross-ratios, harmonic relations, and harmonic maps. For torsion-free modules over noncommutative principal ideal domains, von Staudt–Hua’s theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruples are calculated.     

Exponentially harmonic maps and their properties

We show that the associated quadratic differentials of exponentially harmonic maps are holomorphic under certain circumstance. We study the sufficient and necessary conditions for axially symmetric maps which are exponentially harmonic. We investigate exponentially harmonic equations for rotationally symmetric maps between rotationally symmetric manifolds of low dimensions.     

The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

A construction of conformal-harmonic maps

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells–Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky–Viaclovsky and Lamm.     

Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations

In this paper, we consider the harmonic maps with potential from compact Riemannian manifold with boundary into a convex ball in any Riemannian manifold. We will establish some general properties such as the maximum principles, uniqueness and existence for these maps, and as an application of them, we derive existence and uniqueness result for the Dirichlet problem of the Landau-Lifshitz equations. Received: December 10, 1997 / Accepted: June 29, 1998     

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 its $\epsilon$-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.     

一类调和映射的边界正则性

讨论一类映入球面的满足拟单调不等式的弱调和映射的边界正则性。利用函数的延拓技巧以及Hardy空间和BMO空间的对偶性，对这类弱调和映射的边界正则性给出一个简明的证明。     

A REMARK ON THE UNIQUENESS OF THE HARMONIC MAPS

This paper studies the uniqueness of the harmonic maps from R~n into S~(n-1). At first it is proved that the map ψ(x)=x|x|:B~n→S~(n-1) is the unique energy minimizing harmonic map. Then the uniqueness of the harmonic maps with identity boundary value from B~n to S~(n-1) follows.