A structure theorem of Dirac-harmonic maps between spheres

Harmonic Functions On Manifolds Whose Large Sphere Are Small

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.

Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product $M\times\mathbb{R}_1,$ where $M$ is an arbitrary $n$-dimensional compact Riemannian manifold, $n\geq 2.$ In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once punctured sphere and no harmonic…

k-Harmonic immersion and submersion into a sphere

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k" of k-harmonic. And we also consider k-harmonic submersion, and genelarize Oniciuc's results. Futhermore we construct non harmonic k-harmonic by Hopf map.

Index and nullity of a family of harmonic tori in the sphere

In this note we investigate the $(E)$-index and the $(E)$-nullity of vacuum solutions from the two-torus into the two sphere.

Are all Dirac-harmonic maps uncoupled?

Dirac-harmonic maps $(f,ϕ)$ consist of a map $f:M\to N$ and a twisted spinor $ϕ\inΓ(ΣM\otimes f^*TN)$ and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called \emph{uncoupled}, if $f$ is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass type representations

Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on $S^2$.

Harmonic morphisms on conformally flat 3-spheres

We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.

New examples of r-harmonic immersions into the sphere

Polyharmonic, or $r$-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper $r$-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion $i: S^{n-1}(R)\to S^n$ is a proper $r$-harmonic submanifold of $S^n$ if and only if the radius $R$ is equal to $1/ \sqrt{r}$. We shall also prove the existence of proper $r$-harmonic generalized Clifford's tori into the sphere.

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

It is proved some results about existence and non existence of unit normal sections of submanifolds of the Euclidean space and sphere which associated Gauss maps are harmonic. Some applications to CMC hypersurfaces of the sphere and isoparametric submanifolds are obtained too.

Harmonic Hopf Constructions Between Spheres II

We prove that a necessary condition for the existence of the remaining problem in the harmonic Hopf construction is also sufficient. We also give some topological applications based on our result.

拟调和球的Liouville定理

Liouville theorem for quasi-harmonic sphere

本文证明：若紧致黎曼流形的万有覆盖存在凸函数，则该流形不存在拟调和球，且无需全局增长条件。

球面上共形 Dirac-Einstein 问题的奇异解

Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere

本文研究了共形 Dirac-Einstein 系统的奇异解存在性，利用其共形不变性，在三维球面上构造了一族恰好有两个奇点的奇异解。

Legendre surfaces with harmonic mean curvature vector field in the unit 5-sphere

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Sequences of harmonic maps in the 3-sphere

We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3-sphere, $H$-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold $S^3\times S^3$. As a consequence we can construct sequences of $H$-surfaces and almost complex surfaces.

Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifolds $M_1\not=S^3$ and $M_2\not=S^3,\R¶^3$, which must develop finite time singularity.

Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length

Submission was withdrawn by authors. arXiv:math/0305140

A note on the spectral deformation of harmonic maps from the two-sphere into the unitary group

In [5], together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from $2$-spheres to the unitary group $U(n)$ in terms of freely chosen meromorphic functions on $S^2$. The simplest harmonic maps are the isotropic ones. Using Morse theory Burstall and Guest [1] showed that the harmonic maps come in clusters labeled by the isotropic ones. In this work, using the formula for harmonic maps aforementioned, we describe explicitly this procedure, showing how all harmonic maps can be built from the isotropic ones. [1] F.~E.\ Burstall and M.~A.\ Guest, \textit{Harm…

Construction of Willmore two-spheres via harmonic maps into $SO^+(1,n+3)/(SO^+(1,1)\times SO(n+2))$

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by Hélein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valued meromorphic 1-forms) of totally isotropic Willmore two-spheres in terms of the isotropic harmonic maps. In particular, the corresponding isotropic harmonic maps are of finite uniton type. The proof also contains a concrete way to construct examples of totally isotropic Willmor…

Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.

Nonexistence of quasi-harmonic sphere with large energy

Nonexistence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let $(N,h)$ be a complete noncompact Riemannian manifolds. Assume the universal covering of $(N,h)$ admits a nonnegative strictly convex function with polynomial growth. Then there is no quasi-harmonic spheres $u:\mathbb{R}^n\ra N$ such that $$\lim_{r\ra\infty}r^ne^{-\f{r^2}{4}}\int_{|x|\leq r}e^{-\f{|x|^2}{4}}|\nabla u|^2dx=0.$$ This generalizes a result of the first named author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus very simpl…

Harmonic maps and Kaluza-Klein metrics on spheres

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Finally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.

Harmonic branched coverings and uniformization of CAT($k$) spheres

Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.

Explicit construction of harmonic two-spheres into the complex Grassmannian

We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map $Q$ and a collection of meromorphic functions and their derivatives, we show how to algebraically construct all harmonic maps from the two-sphere into a given Grassmannian $G_p(\mathbb C^n)$. In this setting the uniton number depends on $Q$ and $p$ and we obtain a sharp estimate for it.

Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length

We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N$ is a manifold admitting Killing spinors.

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

For a sequence of coupled fields $\{(ϕ_n,ψ_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.

Harmonic maps and sections on spheres

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector …

The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres

The Weierstrass representation for spheres in $\R^3$ and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or of a cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them a lower estimate of the Willmore …

On the Dirac spectrum of homogeneous 3-spheres

We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$ which are isospectral for the associated classical Dirac operator $D$ must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of $D^2$. We show analogous results for left-invariant metrics on $\operatorname{SO}(3)=S^3/\{\pm1\}$ for each of its two spin structures.

Harmonicity of sections of sphere bundles

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with <,>. This leads to the construction of a lot of new examples of differential forms which are harmonic sections or determine a harmonic map from (M,<,>) into its …

球面$S^4$和双曲空间$H^4$上的新极小曲面 via 谐映射

New Minimal Surfaces of the Sphere $S^4$ and the Hyperbolic Space $H^4$ via Harmonic Morphisms

本文提出通过复值谐映射理论构造偶维球面和双曲空间中二重余维数极小子流形的新方法，首次给出了在$S^4$和$H^4$上显式例子。

A note on the nonexistence of quasi-harmonic spheres

In this paper we study the properties of quasi-harmonic spheres from $\R^m, m>2$. We show that if the universal covering $\tilde N$ of $N$ admits a nonnegative strictly convex function $ρ$ with the exponential growth condition $ρ(y)\leq C\exp\left(\frac14\tilde d(y)^{2/m}\right)$ where $\tilde d(y)$ is the distance function on $\tilde N$, then $N$ does not admit a quasi-harmonic sphere, which generalize Li-Zhu's result \cite{Li2010non}. We also show that if $u$ is a quasi-harmonic sphere, then the property that $u$ is of finite energy ($\int_{\R^m}e(u)e^{-\abs{x}^2/4}\dif x<\infty$) is equival…

Harmonic maps from surfaces of arbitrary genus into spheres

We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from $S^1$) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of ha…

Jacobi fields along harmonic 2-spheres in ${\bf C}P^2$ are integrable

We show that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). This provides one of the few known answers to this problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bf C}P^2$ thus giving the nullity of any such harmonic map; it also has bearing on the behaviour of weakly harmonic $E$-minimizing maps from a 3-manifold…

A characterization of Dirac morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

Regularity of Dirac-harmonic maps

For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $ΣM$, and any compact Riemannian manifold $N$, we show an $ε$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $ψ\in W^{1,p}$ for some $p>2n/3$, th…

Eigenvalue Estimates of the ${\rm spin}^c$ Dirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein man…

球面上的多项式调和态射与特征族

Polynomial harmonic morphisms and eigenfamilies on spheres

本文刻画并构造球面 S^m 上由同次齐次多项式调和态射构成的整体特征族，分析非全纯情形，并在低维中分类二次诱导映射对。

F-Harmonic maps as global maxima

In this note, we show that some F-harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in [15] and [17] for harmonic and p-harmonic maps.

Harmonic Spheres in the Hilbert-Schmidt Grassmannian

We give the twistor description of harmonic maps of the Riemann sphere into the Hilbert-Schmidt Grassmannian. The study of such maps is motivated by the harmonic spheres conjecture formulated in the beginning of this paper.

On the fundamental group of the space of harmonic $2$-spheres in the $n$-sphere

We compute the fundamental group of the "moduli space" of classical solutions of the two dimensional Euclidean $S^n$-model.

Regularity for Dirac-harmonic maps into certain pseudo-Riemannian manifolds

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

Magnetic Dirac-harmonic maps

We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physics. In two dimensions it is conformally invariant. We call critical points of this functional magnetic Dirac-harmonic maps. We study geometric and analytic properties of magnetic Dirac-harmonic maps including their regularity and the removal of isolated singularities.

Some examples of Dirac-harmonic maps

We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension $1$ with target spaces of constant sectional curvature.Before the present article, it remained unclear when the conditions of the theorems in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009…

Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

Regularity at the free boundary for Dirac-harmonic maps from surfaces

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $ε$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.

Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres

We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.

球面间双调和多项式映射的新构造

New constructions of biharmonic polynomial maps between spheres

本文研究球面间由两个齐次多项式映射构成的对角映射，通过计算双张力场，提出了利用不同次数调和齐次多项式映射构造真双调和映射及乘积映射的新方法。

具有最低可能指标的球面调和映射及其性质

Harmonic maps from spheres with lowest possible index and their properties

本文证明球面到复射影空间的低指标调和映射的伪水平弱共形性及奇偶性限制，并给出反例说明结论不适用于格拉斯曼流形。

Dirac-Harmonic Maps

We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|dϕ|^2+(ψ,\Dψ)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

Regularity Theorems and Energy Identities for Dirac-Harmonic Maps

We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from a Riemann surface to a sphere $§^n$. We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.

Harmonic two-spheres in compact symmetric spaces, revisited

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.

Some Notes on Dirac operators on the ${\rm S}^3$ and ${\rm S}^2$ spheres

We describe both the Hodge - de Rham and the spin manifold Dirac operator on the spheres ${\rm S}^3$ and ${\rm S}^2$, following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.

度量空间中能量最小化的调和2-球面

Energy minimizing harmonic 2-spheres in metric spaces

本文提出一种概念简单的度量方法，将Sacks-Uhlenbeck关于闭黎曼流形中非平凡调和2-球面存在性的经典结果推广到一大类紧度量空间。

Some aspects of Dirac-harmonic maps with curvature term

We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces.