The geometry of $Φ_{(3)}$-harmonic maps

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, the $Φ_{(3)}$-conservation law, a monotonicity formula, 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 for $i=1,2,3$, every compact $Φ_{(i)}$-$\operatorname{SSU}$ manifold is $Φ_{(i)}$-$\operatorname{SU}$, and hence is $Φ_{(i)}$-$\operatorname{U}$ (cf. Theorem 9.3). 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. [21, 40, 42, 41, 12, 13]).