The Geometry of $\Phi _S$-Harmonic Maps

Abstract

Just as we study energy density e(u), energy E(u) of harmonic maps, extend them in Wei (An extrinsic average variational method, American Mathematical Society, Providence, 1989) to $\Phi$-energy density $e_{\Phi }(u)$, $\Phi$-energy $E_{\Phi }(u)$ of a map, and $\Phi$-harmonic map $\big ($from the view point of the second symmetric function $\sigma _2$ of a pullback (0, 2)-tensor$\big )$, in this paper, we introduce the notions of the $\Phi _S$-energy density $e_{\Phi _S}(u)$, $\Phi _S$-energy $E_{\Phi _S}(u)$ of a map $u: M \rightarrow N\, ,$ $\Phi _S$-harmonic map, stable $\Phi _S$-harmonic map, and unstable $\Phi _S$-harmonic map, that are associated with the stress-energy tensor S as discussed in (4). We investigate $\Phi _S$-harmonic maps or stress-energy stationary maps between Riemannian manifolds. Liouville type theorems for $\Phi _S$-harmonic maps from complete Riemannian manifolds are established under some 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 (Han and Wei in $\Phi$-harmonic maps and $\Phi$-superstrongly unstable manifolds, 2019), we prove that any stable $\Phi _S$-harmonic map from or to a compact $\Phi$-SSU manifold (to or from any compact manifold) must be constant (cf. Theorems 6.1 and 7.1). We further prove that the homotopic class of any map from any compact manifold into a compact $\Phi$-SSU manifold contains elements of arbitrarily small $\Phi _S$-energy, and the homotopic class of any map from a compact $\Phi$-SSU manifold into any manifold contains elements of arbitrarily small $\Phi _S$-energy (cf. Theorems 8.1 and 9.1). As immediate consequences, we give a simple and direct proof of Theorems 6.1 and 7.1. These Theorems 6.1, 7.1, 8.1 and 9.1 give rise to the concept of $\Phi _S$-strongly unstable $(\Phi _S$-SU) manifolds, extending the notions of strongly unstable $({\text {SU}})$, p-strongly unstable (p-SU), $\Phi$-strongly unstable $(\Phi$-SU) manifolds (cf. [17, 19, 30, 32]). We also introduce the concepts of $\Phi _S$-superstrongly unstable $(\Phi _S$-${\text {SSU}})$ manifold, $\Phi _S$-unstable $(\Phi _S$-${\text {U}})$ manifold and establish a link of $\Phi _S$-${\text {SSU}}$ manifold to p-SSU manifold and topology. Compact $\Phi _S$-${\text {SSU}}$ homogeneous spaces are also studied.