Liouville type theorems and stability of ΦS, p-harmonic maps. (November 2021) - Feng, Shuxiang; Han, Yingbo; Wei, Shihshu Walter
Abstract: In this paper, we first raise the following question: can we obtain the p -stress energy tensor S p that is associated with the p -energy functional E p vanishes under some interesting conditions? This motivates us to introduce the notions of the Φ S, p -energy density e Φ S, p ( u ), and the Φ S, p -energy functional E Φ S, p ( u ) of a map u : M → N, that are related to the p -stress energy tensor S p 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 Φ S, p -energy functional E Φ S, p ( u ) . We also introduce the stress energy tensor S Φ S, p for the Φ S, p -energy functional E Φ S, p, the notions of Φ S, p -harmonic maps, and stable Φ S, p -harmonic maps between Riemannian manifolds. Then we obtain some properties for weakly conformal Φ S, p -harmonic maps and horizontally conformal Φ S, p -harmonic maps, and prove some Liouville type results for Φ S, 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 Φ S, p -SSU manifolds and prove that any stable Φ S, p -harmonic map from or into a compact Φ S, 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 Φ S, p -SSU manifold contains elements of arbitrarily small Φ S, p -energy, and the homotopic class of any map from a compact Φ S, p -SSU manifold into a manifold contains elements of arbitrarily small Φ S, 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 Φ S, p -strongly unstable ( Φ S, p -SU) manifolds, extending the notions of strongly unstable (SU), p -strongly unstable ( p -SU), Φ -strongly unstable ( Φ -SU) and Φ S -strongly unstable ( Φ S -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 Φ S superstrongly unstable ( Φ S -SSU) manifolds are strongly unstable (SU), p -strongly unstable ( p -SU), Φ -strongly unstable ( Φ -SU) and Φ S -strongly unstable ( Φ S -SU) manifolds respectively, and enjoy their wonderful properties. We also introduce the concepts of Φ S, p -unstable ( Φ S, p -U) manifold and establish a link of Φ S, p -SSU manifold to p -SSU manifold and topology. Compact Φ S, p -SSU homogeneous spaces are studied.