The Geometry of $$\Phi _S$$-Harmonic Maps

Abstract

In this paper, we motivate and extend the study of harmonic maps or Φ (1) -harmonic maps (cf [15], Remark 1.3 (iii)), Φ-harmonic maps or Φ (2) -harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of Φ (3) -harmonic maps by unified geometric analytic methods. We define the notion of Φ (3) -harmonic maps and obtain the first variation formula and the second variation formula of the Φ (3) -energy functional E Φ (3) . By using a stress-energy tensor and the asymptotic assumption of maps at infinity, we prove Liouville type results for Φ (3) -harmonic maps. We introduce the notion of Φ (3) -Superstrongly Unstable (Φ (3) -SSU) manifold and provide many interesting examples. By using an extrinsic average variational method in the calculus of variations (cf. [51,49]), we find Φ (3) -SSU manifold and prove that any stable Φ (3) -harmonic maps from a compact Φ (3) -SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ (3) -SSU manifold must be constant. We also prove that the homotopy class of any map from a compact Φ (3) -SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ (3) -SSU manifold contains elements of arbitrarily small Φ (3) -energy. We call a compact Riemannian manifold M to be Φ (3) -strongly unstable (Φ (3) -SU) if it is not the target or domain of a nonconstant stable Φ (3) -harmonic map (from or into any compact Riemannian manifold) and also the homotopy class of any map to or from M (from or into any compact Riemannian manifold) contains elements of arbitrarily small Φ (3) -energy. We prove that every compact Φ (3) -SSU manifold is Φ (3) -SU. As consequences, we obtain topological vanishing theorems and sphere theorems by employing a Φ (3) -harmoic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic ([45]), minimal surface, stable rectifiable current ([34, 29, 50]), p-harmonic map (cf. [53]), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or Φ (1) -harmonic maps, p-harmonic maps, Φ S -harmonic maps, Φ S,p -harmonic maps, Φ (2) -harmonic maps, etc., (cf.