In our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite-dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-) Riemannian geodesics and constant mean curvature hypersurfaces.

In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where processes occur on surfaces or thin layers. We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface. The main ingredients of MsFEM are (1) the construction of multiscale basis functions and (2) a global coupling of these basis functions. For the construction of multiscale basis functions, our approach uses the transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved along the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration. We discuss the use of appropriate transformation operators that improve the accuracy of the method. The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition. In this paper, we consider such transformations based on harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied Math., LX(2007), pp. 675-723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two types of deformations are used for multiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results show that one can improve the accuracy of the simulations when a global information is used.

Guest–Ohnita and Crawford have shown the path-connectedness of the space of harmonic maps from ${{S}^{2}}$ to $\text{C}{{P}^{n}}$ of a fixed degree and energy. It is well known that the $\partial$ transform is defined on this space. In this paper, we will show that the space is decomposed into mutually disjoint connected subspaces on which $\partial$ is homeomorphic.

We prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic.

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions.

We study the harmonicity of maps to or from cosymplectic manifolds by relating them to maps to or from Kähler spaces.

We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac{1}{2}\int_{B^2}|\nabla u|^2 {\rm d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.

Let M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.