Bounded harmonic maps on a class of manifolds

Abstract

Abstract. Without imposing any curvature assumptions, we show that bounded harmonic maps with image contained in a regular geodesic ball share similar behaviour at infinity with the bounded harmonic functions on the domain manifold. §0In this note, among other things we will prove a Liouville type theorem on harmonic maps. It was first proved by Yau [Y] that any positive harmonic function on a complete noncompact manifold with nonnegative Ricci curvature must be constant. Later, Cheng [Cg] proved that any harmonic map with bounded image from a complete noncompact manifold with nonnegative Ricci curvature into a Cartan-Hadamard manifold must be a constant map. Using a similar method, Choi [Ch] was able to generalize Cheng's result. He showed that any harmonic map from a complete noncompact manifold with nonnegative Ricci curvature into a complete manifold with sectional curvature bounded from above by K > 0 is a constant map, provided that the image of the harmonic map lies inside a regular geodesic ball (see Definition 1.3). It was proved by Kendall [Ke] that the result is still true by only assuming the domain manifold supports no nonconstant bounded harmonic functions. If we relax the condition that the manifold has nonnegative Ricci curvature outside a compact set, then the theorem of Yau [Y] is no longer true. In fact, the behavior of bounded and positive harmonic functions on a complete noncompact manifold with nonnegative sectional curvature outside a compact set has been studied thoroughly in [L-T 1]. One of the results in [L-T 1] is that if M has nonnegative sectional curvature outside a compact set, then M has finitely many ends (see §1), and a bounded harmonic function defined near the infinity on an end will be asymptotically constant. There are many kinds of manifolds which satisfy the same property. See the examples in §1. One of the main results we obtain in this note is that if every bounded harmonic function defined near infinity of an end of a manifold M with respect to some compact set is asymptotically constant, then every bounded harmonic map from M into a regular ball of another manifold is also asymptotically constant at the infinity of each end. There is no curvature assumption on M . Moreover, such a map will have finite total energy.