We study the stability of harmonic morphisms as a subclass of harmonic maps. As a general result we show that any harmonic morphism to a manifold of dimension at least three is stable with respect to some Riemannian metric on the target. Furthermore we link the index and nullity of the composition of harmonic morphisms with the index and nullity of the composed 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 ǫ-regularity theorem for weakly Dirac-harmonic maps (φ, ψ) : M ⊗ΣM → N ⊗ φ∗TN . 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 establi...

The surface-matching problem is investigated in this paper using a mathematical tool called harmonic maps. The theory of harmonic maps studies the mapping between different metric manifolds from the energyminimization point of view. With the application of harmonic maps, a surface representation called harmonic shape images is generated to represent and match 3D freeform surfaces. The basic ide...

Harmonic maps are mappings between Riemannian manifolds which extremize a natural energy functional. They have been studied for many years in differential geometry, and in particle physics as nonlinear sigma models. We shall report on recent progress in understanding their infinitesimal deformations, the so-called Jacobi fields. It is important to know whether the Jacobi fields along harmonic m...

§1 Background and Setup Let M be an m-dimensional, compact, Riemannian manifold endowed with the metric dsM = gij dx i dx , where {x, x, · · · , x} is a local coordinate system of M. Suppose N is an n-dimensional, complete, Riemannian manifold with metric given by dsN = hαβ du α du , where {u, u, · · · , u} is a local coordinate system of N. Let f : M → N be a C mapping from M into N . Definiti...

(1.1) M' is complete and its sectional curvatures are non-positive. In terms of local coordinates x = (x, . . . , x) on M and y = (y, . . . , y) on M', let the respective Riemann elements of arc-length be ds = gij dx dx\ ds' = g'a$ dy a dy& and r^-fc, T'Vy be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M') or its coordina...

This article surveys research on the existence, structure, behavior, and asymptotics of singularities of harmonic maps.