Recently there has been much interest in the Liouville type theorems for harmonic maps. For a detailed survey and progress in this direction, see the works by Hildebrandt [4], Eells and Lemaire [2]. Here we would like to mention that for all known results, the conditions on the harmonic maps can be divided into two kinds. The first of these conditions concerns the finiteness of the energy of th...

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be ...

Let X be a complete noncompact Riemannian manifold with Ricci curvature and Sobolev radius (see §6 for the definition) bounded from below and Y a complete Riemannian manifold with nonpositive sectional curvature. We shall study some situations where a smooth map f : X → Y can be deformed continuously into a harmonic map, using a naturally defined flow. The flow used here is not the usual harmon...

In this paper, we study harmonic maps relative to α-connections, but not necessarily standard harmonic maps. A standard harmonic map is defined by the first variation of the energy functional of a map. A harmonic map relative to an α-connection is defined by an equation similar to a first variational equation, though it is not induced by the first variation of the standard energy functional. In...

Let F : M → N be a harmonic map between complete Riemannian manifolds. Assume that N is simply connected with sectional curvature bounded between two negative constants. If F is a quasiconformal harmonic diffeomorphism, then M supports an infinite dimensional space of bounded harmonic functions. On the other hand, if M supports no non-constant bounded harmonic functions, then any harmonic map o...

A smooth map f: M ! N between semi-riemannian manifolds is called a harmonic morphism if f pulls back harmonic functions (i.e., local solutions of the Laplace{Beltrami equation) on N into harmonic functions on M. It is shown that a harmonic morphism is the same as a harmonic map which is moreover horizontally weakly conformal, these two notions being likewise carried over from the riemannian ca...