Harmonic maps with finite total energy

Abstract

We will give a criteria for a nonnegative subharmonic function with finite energy on a complete manifold to be bounded. Using this we will prove that if on a complete noncompact Riemannian manifold M M , every harmonic function with finite energy is bounded, then every harmonic map with finite total energy from M M into a Cartan-Hadamard manifold must also have bounded image. No assumption on the curvature of M M is required. As a consequence, we will generalize some of the uniqueness results on homotopic harmonic maps by Schoen and Yau.