The geometry of [formula omitted]-harmonic maps

Abstract

In this paper, we motivate and extend the study of harmonic maps or Φ(1)-harmonic maps (cf. Eells and Sampson, 1964, Remark 1.3 (iii)), Φ-harmonic maps or Φ(2)-harmonic maps (cf. Han and Wei, 2022, 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 EΦ(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. Wei, 1998; Wei, 1984), 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)-harmonic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic (Synge, 1936), minimal surface, stable rectifiable current (Lawson Jr. and Simons, 1973; Howard and Wei, 2015; Wei, 1984), p-harmonic map (cf. Wei, 1998), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or Φ(1)-harmonic maps, p-harmonic maps, ΦS-harmonic maps, ΦS,p-harmonic maps, Φ(2)-harmonic maps, etc., (cf. Howard and Wei, 1986; Wei, 1992; Wei and Yau, 1994; Wei, 1998; Feng-Han-Li-Wei, 2021; Feng-Han-Wei, 2021).