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.

正如我们研究谐波图的能量密度e（u），能量E（u）一样，将它们在Wei（一种非平均平均变分方法，美国数学协会，普罗维登斯，1989年）中扩展为\（\ Phi \）-能量密度\ （e _ {\ Phi}（u）\），\（\ Phi \）-能量\（E _ {\ Phi}（u）\）和地图（\（\ Phi \）-谐波地图\（\ big （\）从回拉（0，2）-张量\（\ big）\）的第二个对称函数\（\ sigma _2 \）的角度出发，在本文中，我们介绍\（\ hi_S \）-能量密度\（e _ {\ Phi _S}（u）\），\（\ Phi _S \）-能量\（E _ {\ Phi _S}（u）\）地图\（u：M \ rightarrow N \，， \） \（\ Phi _S \）-调和图，稳定\（\ Phi _S \）-调和图和不稳定\（\ Phi _S \）-调和图，与所讨论的应力-能量张量S相关在（4）中。我们研究黎曼流形之间的\（\ Phi _S \）-调和图或应力能平稳图。Liouville定理\（\ Phi _S \）在某些条件下，在距离函数的Hessian和该图在无穷远处的渐近行为的条件下，建立了完整黎曼流形的高次谐波图。通过外部微分变分法（在\（\ Phi \）-调和映射和\（\ Phi \）-超强不稳定流形中的Han和Wei ，2019），我们证明了任何稳定的\（\ Phi _S \） -从紧凑型\（\ Phi \）- SSU流形（到或来自任何紧凑型流形）的谐波映射必须是恒定的（请参见定理6.1和7.1）。我们进一步证明，从任何紧流形到紧\（\ Phi \）- SSU流形的任何映射的同构类都包含任意小的\（\ Phi _S \）元素-能量，以及从紧凑的\（\ Phi \）- SSU流形到任何流形的任何映射的同构类，包含任意小的\（\ Phi _S \）-能量的元素（参见定理8.1和9.1）。作为直接后果，我们给出定理6.1和7.1的简单直接证明。这些定理6.1、7.1、8.1和9.1产生了\（\ Phi _S \）-高度不稳定\（（（Phi _S \）- SU）流形的概念，扩展了高度不稳定\（（{ {SU}}）\），p-极不稳定（p -SU），\（\ Phi \）-极不稳定\（（\ Phi \）- SU）流形（参见[17，19，30，32] ）。我们还介绍了\（\ Phi _S \）-非常不稳定\（（\ Phi _S \） - \（{\ text {SSU}}）\）歧管，\（\ Phi _S \）-不稳定\（（（Phi _S \） - \（{\ text {U}}）\）流形，并建立\（\ Phi _S \） - \（{\ text {SSU}} \\}流形到p -SSU流形和拓扑的链接。紧凑\（\ Phi _S \） - \（{\ text {SSU}} \）齐次空间也被研究。