A Liouville theorem for F-harmonic maps with finite F-energy

Abstract

Let (M, g) be a m-dimensional complete Riemannian manifold with a pole, and (N, h) a Riemannian manifold. Let F : ℝ⁺ → ℝ⁺ be a strictly increasing C2 function such that F(0) = 0 and dF := sup(tF′ (t)) (F(t))1) < ∞. We show that if dF < m/2, then every F-harmonic map u : M → N with finite F-energy (i.e. a local extremal of EF(u) := ∫M F(|du|2 /2) dVg and EF(u) is finite) is a constant map provided that the radial curvature of M satisfies a pinching condition depending to dF.