We motivate and define \(\Phi \)-energy density, \(\Phi \)-energy, \(\Phi \)-harmonic maps and stable \(\Phi \)-harmonic maps. Whereas harmonic maps or p-harmonic maps can be viewed as critical points of the integral of the first symmetric function \(\sigma _1\) of a pull-back tensor, \(\Phi \)-harmonic maps can be viewed as critical points of the integral of the second symmetric function \(\sigma _2\) of a pull-back tensor. By an extrinsic average variational method in the calculus of variations [cf. Howard and Wei (Trans Am Math Soc 294:319–331, 1986), Wei and Yau (J Geom Anal 4(2):247–272, 1994), Wei (Indiana Univ Math J 47(2):625–670, 1998) and Howard and Wei (Contemp Math 646:127–167, 2015)], we derive the average second variation formulas for \(\Phi \)-energy functional, express them in orthogonal notation in terms of the differential matrix, and find \(\Phi \)-superstrongly unstable \((\Phi \)-\(\text {SSU})\) manifolds. We prove, in particular that every compact \(\Phi \)-\(\text {SSU}\) manifold must be \(\Phi \)-strongly unstable \((\Phi \)-\(\text {SU})\), i.e., (a) A compact \(\Phi \)-\(\text {SSU}\) manifold cannot be the target of any nonconstant stable \(\Phi \)-harmonic maps from any manifold, (b) The homotopic class of any map from any manifold into a compact \(\Phi \)-\(\text {SSU}\) manifold contains elements of arbitrarily small \(\Phi \)-energy, (c) A compact \(\Phi \)-\(\text {SSU}\) manifold cannot be the domain of any nonconstant stable \(\Phi \)-harmonic map into any manifold, and (d) The homotopic class of any map from a compact \(\Phi \)-\(\text {SSU}\) manifold into any manifold contains elements of arbitrarily small \(\Phi \)-energy [cf. Theorem 1.1(a),(b),(c), and (d).] We provide many examples of \(\Phi \)-\(\text {SSU}\) manifolds, which include but not limit to spheres or some unstable Yang-Mills fields [cf. Bourguignon et al. (Proc Natl Acad Sci 76(4):1550–1553, 1979), Bourguignon and Lawson (Commun Math Phys 79(2):189–230, 1981), Kobayashi et al. (Math Z 193(2):165–189, 1986), Wei (Indiana Univ Math J 33(4):511–529, 1984) and Wu et al. (Br J Math Comput Sci 8(4):318–329, 2015)], and examples of \(\Phi \)-harmonic, or \(\Phi \)-unstable map from or into \(\Phi \)-\(\text {SSU}\) manifold that are not constant. We establish a link of \(\Phi \)-SSU manifold to p-SSU manifold and topology. The extrinsic average variational method in the calculus of variations, employed is in contrast to an average method in PDE that we applied in Chen and Wei (J Geom Symmetry Phys 52:27–46, 2019) to obtain sharp growth estimates for warping functions in multiply warped product manifolds.

我们激励并定义了\(\Phi \) -能量密度、\(\Phi \) -能量、\(\Phi \) -谐波映射和稳定的\(\Phi \) -谐波映射。而谐波映射或p谐波映射可以看作是回拉张量的第一个对称函数\(\sigma _1\)积分的临界点，而\(\Phi \)谐波映射可以看作是临界点第二个对称函数的积分点\(\sigma _2\)一个回拉张量。通过变分计算中的外在平均变分方法[cf. Howard and Wei (Trans Am Math Soc 294:319–331, 1986), Wei and Yau (J Geom Anal 4(2):247–272, 1994), Wei (Indiana Univ Math J 47(2):625–670 , 1998) 以及 Howard 和 Wei (Contemp Math 646:127–167, 2015)]，我们推导出了\(\Phi \) -energy 泛函的平均二阶变分公式，用微分矩阵的正交符号表示它们，并找到\(\Phi \) -超强不稳定\((\Phi \) - \(\text {SSU})\)流形。我们证明，特别是每个紧致的\(\Phi \) - \(\text {SSU}\)流形必须是\(\Phi \) -强不稳定的\((\Phi \) - \(\text {SU})\)，即， (a) 一个紧凑的\(\Phi \) - \(\text {SSU}\)流形不能是任何非常量的目标稳定的\(\Phi \) -来自任何流形的谐波映射，(b) 从任何流形到紧凑\(\Phi \) - \(\text {SSU}\)流形的任何映射的同伦类包含任意元素小\(\Phi \) -energy, (c) 紧凑\(\Phi \) - \(\text {SSU}\)流形不能是任何非常量稳定\(\Phi \) -谐波映射到任何流形，以及 (d) 来自紧致\(\Phi \)的任何映射的同伦类-\(\text {SSU}\)流形到任何流形中都包含任意小的\(\Phi \) -energy [cf. 定理 1.1(a)、(b)、(c) 和 (d)。] 我们提供了许多\(\Phi \) - \(\text {SSU}\)流形的例子，其中包括但不限于球体或一些不稳定的杨-米尔斯场 [cf. 布吉尼翁等。（Proc Natl Acad Sci 76(4):1550–1553, 1979）、Bourguignon 和 Lawson（Commun Math Phys 79(2):189–230, 1981）、Kobayashi 等人。(Math Z 193(2):165–189, 1986), Wei (Indiana Univ Math J 33(4):511–529, 1984) 和 Wu 等人。(Br J Math Comput Sci 8(4):318–329, 2015)]，以及\(\Phi \) -harmonic 或\(\Phi \) - 从或进入\(\Phi \) 的不稳定映射示例- \(\text {SSU}\)不恒定的流形。我们建立了\(\Phi \) -SSU 流形到p -SSU 流形和拓扑的链接。变分计算中采用的外在平均变分方法与我们在 Chen 和 Wei (J Geom Symmetry Phys 52:27–46, 2019) 中应用的 PDE 中的平均方法形成对比，以获得翘曲函数的急剧增长估计多重扭曲的产品流形。