$Φ$ -调和映射与$Φ$ -超强非稳定流形
$Φ$-Harmonic Maps and $Φ$-Superstrongly Unstable Manifolds
摘要：
本文引入并定义了$Φ$ -能量密度、$Φ$ -能量、$Φ$ -调和映射和稳定的$Φ$ -调和映射.调和映射或$p$ -调和映射可以被视为拉回张量的$σ_1$的积分的临界点，而$Φ$ -调和映射可以被视为拉回张量的$σ_2$的积分的临界点。通过变分法中的外平均变分方法（参见。\cite{HW,WY,13,HaW}），导出了$Φ$ -能量泛函的平均二阶变分公式，并用微分矩阵的正交符号表示，并找到了$Φ$ -超强不稳定$(Φ$ - $\text{SSU})$流形.特别地，我们证明了每个紧$Φ$ - $\text{SSU}$流形必是$Φ$ -强不稳定的$(Φ$ - $\text{SU})$，即，紧$Φ$ - $\text{SSU}$流形不能是来自任何流形的任何非常数稳定$Φ$ -调和映射的目标，$\rm (b)$从任何流形到紧$Φ$ - $\text{SSU}$流形的任何映射的同伦类包含任意小$Φ$ -能量，$(\rm c)$紧$Φ$ - $\text{SSU}$流形不能是到任何流形的任何非常数稳定$Φ$ -调和映射的域，和$(\rm d)$从紧致$Φ$ - $\text{SSU}$流形到任何流形的任何映射的同伦类包含任意小$Φ$ -能量的元素（参见.定理$1.1 (a),(b),(c)$和$(d)$。）给出了$Φ$-SSU流形的许多例子，建立了$Φ$-SSU流形与$p$-SSU流形和拓扑的联系。我们采用的变分法中的外在平均变分方法与我们在\cite {CW}中应用的PDE中的平均方法相反，以获得多个翘曲乘积流形中翘曲函数的急剧增长估计。
英文摘要：
In this paper, we motivate and define $Φ$-energy density, $Φ$-energy, $Φ$-harmonic maps and stable $Φ$-harmonic maps. Whereas harmonic maps or $p$-harmonic maps can be viewed as critical points of the integral of $σ_1$ of a pull-back tensor, $Φ$-harmonic maps can be viewed as critical points of the integral of $σ_2$ of a pull-back tensor. By an extrinsic average variational method in the calculus of variations (cf. \cite{HW,WY,13,HaW}), we derive the average second variation formulas for $Φ$-energy functional, express them in orthogonal notation in terms of the differential matrix, and find $Φ$-superstrongly unstable $(Φ$-$\text{SSU})$ manifolds. We prove, in particular that every compact $Φ$-$\text{SSU}$ manifold must be $Φ$-strongly unstable $(Φ$-$\text{SU})$, i.e., $(a)$ A compact $Φ$-$\text{SSU}$ manifold cannot be the target of any nonconstant stable $Φ$-harmonic maps from any manifold, $\rm (b)$ The homotopic class of any map from any manifold into a compact $Φ$-$\text{SSU}$ manifold contains elements of arbitrarily small $Φ$-energy, $(\rm c)$ A compact $Φ$-$\text{SSU}$ manifold cannot be the domain of any nonconstant stable $Φ$-harmonic map into any manifold, and $(\rm d)$ The homotopic class of any map from a compact $Φ$-$\text{SSU}$ manifold into any manifold contains elements of arbitrarily small $Φ$-energy (cf. Theorem $1.1 (a),(b),(c)$, and $(d)$.) We also provide many examples of $Φ$-SSU manifolds, and establish a link of $Φ$-SSU manifold to $p$-SSU manifold and topology. The extrinsic average variational method in the calculus of variations that we have employed is in contrast to an average method in PDE that we applied in \cite {CW} to obtain sharp growth estimates for warping functions in multiply warped product manifolds.
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