REMARKS OF EXPONENTIALLY HARMONIC MAPS

Let (M, g(0)) be a complete Riemannian manifold with a pole x(0) and (N, h) be a Riemannian manifold. We show that if f : (M, eta(2) g(0)) -> (N, h) is an exponentially harmonic map such that eta (a smooth function on M) satisfies some condition (*), then certain monotonicity formula is derived. We study the monotonicity of exponentially harmonic maps under a few different circumstances and discuss their vanishing.