Harmonic morphisms are smooth maps between Riemannian manifolds which preserve Laplace's equation. They are characterised as harmonic maps which are horizontally weakly conformal 14, 20]. R.L. Bryant 7] proved that there are precisely two types of harmonic morphisms with one-dimensional bres which can be deened on a constant curvature space of dimension at least four. Here we prove that, on an ...

Let Σ be a compact Riemann surface. Any sequence fn : Σ —> M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : Σ -> M and a tree of bubbles fk : S 2 -> M. We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the fn converge pointwise. We then give explicit counterexam...

In this note we examine the heat flow for harmonic maps and Yang-Mills equations in dimensions two and four respectively. We show that these two flows are qualitatively similar for degree-2 harmonic maps.

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics. 2000 Mathematics Subject Classification. Primary 30C85. Secondary 30C65.

We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some important applications of smooth harmonic maps can be obtained in this setting.

In this paper we study the topology of the space of harmonic maps from S to CP. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to CPn for n≥2. We show that the components of maps to CP are complex manifolds.

Harmonic maps are viewed as maps sending a xed diiusion to manifold-valued martingales. Under a convexity condition, we prove that the continuity of real-valued harmonic functions implies the continuity of harmonic maps. Then we prove with a probabilistic method that continuous harmonic maps are smooth under HH ormander's condition; the proof relies on the study of martingales with values in th...

In this paper we establish the higher-dimensional energy bubbling results for harmonic maps to spheres. We have shown in particular that the energy density of concentrations has to be the sum of energies of harmonic maps from the standard 2dimensional spheres. The result also applies to the structure of tangent maps of stationary harmonic maps at either a singularity or infinity. 0. Introductio...

Many topics in integrable surface geometry may be unified by application of the highly developed theory of harmonic maps of surfaces into (pseudo-)Riemannian symmetric spaces. On the one hand, such harmonic maps comprise an integrable system with spectral deformations, algebro-geometric solutions and dressing actions of loop groups generated by Bäcklund transforms [5], [6], [14], [21], [24]. On...