Liouville type theorems and stability of [formula omitted]-harmonic maps

Abstract

In this paper, we first raise the following question: can we obtain the p-stress energy tensor Sp that is associated with the p-energy functional Ep vanishes under some interesting conditions? This motivates us to introduce the notions of the ΦS,p-energy density eΦS,p(u), and the ΦS,p-energy functional EΦS,p(u) of a map u:M→N, that are related to the p-stress energy tensor Sp of a smooth map u between two Riemannian manifolds M and N. We derive the first variation formula of type I and type II, and the second variation formula for the ΦS,p-energy functional EΦS,p(u). We also introduce the stress energy tensor SΦS,p for the ΦS,p-energy functional EΦS,p, the notions of ΦS,p-harmonic maps, and stable ΦS,p-harmonic maps between Riemannian manifolds. Then we obtain some properties for weakly conformal ΦS,p-harmonic maps and horizontally conformal ΦS,p-harmonic maps, and prove some Liouville type results for ΦS,p-harmonic maps from some complete Riemannian manifolds under various conditions on the Hessian of the distance function and the asymptotic behavior of the map at infinity. By an extrinsic average variational method in the calculus of variations (Wei; 1989, 1983), we find ΦS,p-SSU manifolds and prove that any stable ΦS,p-harmonic map from or into a compact ΦS,p-SSU manifold (to or from a compact manifold) must be constant (cf. Theorems 5.1 and 6.1). We further prove that the homotopic class of any map from a compact manifold into a compact ΦS,p-SSU manifold contains elements of arbitrarily small ΦS,p-energy, and the homotopic class of any map from a compact ΦS,p-SSU manifold into a manifold contains elements of arbitrarily small ΦS,p-energy (cf. Theorems 7.1 and 8.2). As immediate consequences, we give a simple and direct proof of the above Theorems 5.1 and 6.1. These Theorems 5.1, 6.1, 7.1 and 8.2 give rise to the concept of ΦS,p-strongly unstable (ΦS,p-SU) manifolds, extending the notions of strongly unstable (SU), p-strongly unstable (p-SU), Φ-strongly unstable (Φ-SU) and ΦS-strongly unstable (ΦS-SU) manifolds (cf. Howard and Wei, 1986; Wei and Yau, 1994; Wei, 1998; Han and Wei, 2019; Feng et al., 2021). Hence, superstrongly unstable (SSU), p-superstrongly unstable (p-SSU), Φ-superstrongly unstable (Φ-SSU) and ΦS superstrongly unstable (ΦS-SSU) manifolds are strongly unstable (SU), p-strongly unstable (p-SU), Φ-strongly unstable (Φ-SU) and ΦS-strongly unstable (ΦS-SU) manifolds respectively, and enjoy their wonderful properties. We also introduce the concepts of ΦS,p-unstable (ΦS,p-U) manifold and establish a link of ΦS,p-SSU manifold to p-SSU manifold and topology. Compact ΦS,p-SSU homogeneous spaces are studied.