LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

Abstract

Abstract. Let M be a complete Riemannian manifold and let N be aRiemannian manifold of non-positive sectional curvature. Assume thatRic M ≥ − 4(p−1)p 2 µ 0 at all x ∈ M and Vol(M) is infinite, where µ 0 > 0 isthe infimum of the spectrum of the Laplacian acting on L 2 -functions onM. Then any p-harmonic map φ : M → N of finite p-energy is constant.Also, we study Liouville type theorem for p-harmonic morphism. 1. IntroductionLet (M,g) and (N,h) be smooth Riemannian manifolds and let φ : M → Nbe a smooth map. For a compact domain Ω ⊂ M, the p-energy E p (φ;Ω) of φover Ω is defined by(1.1) E p (φ;Ω) =1pZ Ω |dφ| p µ M ,where the differential dφ is a section of the bundle T ∗ M ⊗ φ −1 TN → M andφ −1 TN denotes the pull-back bundle via the map φ. The bundle T ∗ M ⊗φ −1 TN → M carries the connection ∇ induced by the Levi-Civita connectionson M and N. A map φ : M → N is called p-harmonic if the p-tension fieldτ p (φ) = 0, which is defined by(1.2) τ p (φ) = tr g ∇(|dφ| p−2 dφ),where tr g denote the trace with respect to the metric g. A p-harmonic map φ isa critical point of the energy functional defined by (1.1) on any compact domainΩ ⊂ M. When p = 2, p-harmonic maps are well-known to be harmonic maps.Several studies are given for harmonic maps (see [5], [6], [7], [8], [10], [11], [12],[13], [14], [16]). Let µ