An energy estimate of an exterior problem and a Liouville theorem for harmonic maps

Abstract

We prove that the exterior harmonic maps, from Rn\\Ω(n ≧ 3) to a bounded strictly convex geodesic ball of some Riemannian manifolds, have finite conformal invariant energy. A consequence of this estimate is a Liouville theorem which states that harmonic maps between Euclidean space (Rn, g0) and Riemannian manifolds are constant maps provided their image at infinity falls into a bounded strictly convex geodesic ball.