Narrow Results

After formulating the notion of integrability for axially symmetric harmonic maps from ${ℝ}^3$ into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding $N$-solitonic harmonic maps into a non-compact Grassmann manifold $SU(p,q)/S(U(p)\times U(q))$ is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases $(p=1,q=1)$ and $(p=2,q=1)$ and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

We obtain lower and upper energy estimates for harmonic maps between Riemannian manifolds under natural curvature conditions leading to various Liouville-type theorems. Some of the methods described may also be applied to vanishing-type problems for vector bundle-valued harmonic forms.

We introduce the class of pseudo-horizontally homothetic maps from a Riemann manifold to a Kähler manifold, and we study some of their properties. For example, we prove that a pseudo-horizontally homothetic submersion is harmonic if and only if it has minimal fibres, and a pseudo-horizontally homothetic harmonic submersion pulls back complex submanifolds into minimal submanifolds.

This paper aims to investigate the geometry of the moduli spaces of harmonic maps of compact Riemann surfaces into compact Lie groups or compact symmetric spaces. The approach here is to study the gauge theoretic equations for such harmonic maps and the moduli space of their solutions. We discuss the S¹-action, the hyper-presymplectic structure, the energy function, the Hitchin map, the flag transforms on the moduli space, several kinds of subspaces in the moduli space, and the relationship among them, especially the structure of the critical point subset for the energy function on the moduli space. As results, we show that every uniton solution is a critical point of the energy function on the moduli space, and moreover we give a characterization of the fixed point subset fixed by S¹-action in terms of a flag transform.

We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere .

If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from or to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation.

We conclude by an example of a non-constant harmonic morphism from to .

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

Let $B_n$ be the open unit ball in ${ℂ}^n$ and let $(M^m,g)$ be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map $u\in C^2(\overline{B_n},M)$ satisfying the boundary condition that ${\sum }_{i,j=1}^n{z}_i{\bar{z}}_j{u}_{i\bar{j}}=0$ on $\partial B_n$. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on $B_n$.

We show that if ${\{M_t\}}_{t\in \mathrm{\Delta }}$ is a polarized family of compact Kähler manifolds over the open unit disk $\mathrm{\Delta }$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if ${\{{\varphi }_t:M_t\to N\}}_{t\in \mathrm{\Delta }}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E({\varphi }_t)$ is a subharmonic function of $t\in \mathrm{\Delta }$.

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called pseudo harmonic morphisms, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Finally, we construct a canonical f-structure associated to every map satisfying (PHWC) and find conditions on this f-structure to ensure the harmonicity of the map.

Let G denote either SO₀(n,1) or SU(n,1). We study restrictions of cohomological representations of G to subgroups H isomorphic to SO₀(m,1) or SU(m,1). We prove that such a restriction contains, as a subrepresentation, some cohomological representation of H. When G = SU(n,1), we extend a result of Delorme by showing that cohomological representations are non-spherical in a strong sense: if H is isomorphic to SU(m,1) (with 1 ≤ m ≤ n), then the restriction of a cohomological representation to H is non-spherical.

We consider variational equations from micromagnetics for thin films of ferromagnetic materials and study solutions as the thickness of the film tends to 0. We find a behavior which is similar to that of sequences of harmonic maps in two dimensions.

We analyze an elastic surface energy which was recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. We show how a novel approach in modeling the surface’s extrinsic geometry leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: (i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; (ii) we prove, by a Ginzburg–Landau approximation, the well-posedness of the gradient flow of the energy; (iii) in the case of a parametrized torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

We study nonlinear $\sigma$-models defined on a noncommutative torus as a two-dimensional string worldsheet. We consider (i) a two-point space, (ii) a circle, (iii) a noncommutative torus, (iv) a classical group $SU(2,ℂ)$ as examples of space-time. Based on established results, the trivial harmonic unitaries of the noncommutative chiral model known as local minima are shown not to be global minima by comparing them to the symmetric unitaries derived from instanton solutions of the noncommutative Ising model corresponding to a two-point space. In addition, a ${\mathbb{Z}}^2$-action on field maps is introduced to a noncommutative torus, and its action on solutions of various Euler–Lagrange equations is described.

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

Using the properties of Cartan tensor, we rewrite the second variation formula for harmonic maps between Finsler manifolds, and we prove that there is no non-degenerate stable harmonic map from a compact SSU manifold to any Finsler manifold, which is obtained by Howard and Wei for the Riemannian case. We also include a proof of a theorem of Shen–Wei which states that there is no non-degenerate stable harmonic map from a compact Finsler manifold to any SSU manifold, by the same second variational formula (see Eq. (2.1) in [Y. B. Shen and S. W. Wei, The stability of harmonic maps on Finster manifolds, Houston J. Math. 34 (2008) 1049–1056]) and the same method [S. W. Wei, An extrinsic average variational method, in Recent Developments in Geometry, Contemporary Mathematics, Vol. 101 (American Mathematical Society, Providence, RI, 1989), pp. 55–78].

We give an explicit representation formula for the p-energy of Sobolev maps with values in a metric space as defined by Korevaar and Schoen [Comm. Anal. Geom.1(3–4) (1993) 561–659]. The formula is written in terms of the Lipschitz compositions introduced by Ambrosio [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)3(17) (1990) 439–478], thus further relating the two different definitions considered in the literature.