$\Phi$-Harmonic Maps and $\Phi$-Superstrongly Unstable Manifolds

TL;DR

This paper introduces $\Phi$-harmonic maps and $\Phi$-superstrongly unstable manifolds, deriving second variation formulas and establishing their instability properties, with implications for the existence and energy of maps between such manifolds.

Contribution

It defines $\Phi$-harmonic maps and $\Phi$-SSU manifolds, derives their second variation formulas, and proves their instability properties, linking topology and variational methods.

Findings

$\Phi$-SSU$ manifolds cannot be targets of stable $\Phi$-harmonic maps.

Homotopic classes into $\Phi$-SSU$ manifolds contain maps with arbitrarily small $\Phi$-energy.

$\Phi$-SSU$ manifolds are strongly unstable in the $\Phi$-harmonic map setting.

Abstract

In this paper, 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 $\sigma_1$ of a pull-back tensor, $\Phi$-harmonic maps can be viewed as critical points of the integral of $\sigma_2$ 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 $\Phi$-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, $\rm (b)$ The homotopic class of any map from any manifold into a compact $\Phi$-$\text{SSU}$ manifold contains elements of arbitrarily small $\Phi$-energy, $(\rm c)$ A compact $\Phi$-$\text{SSU}$ manifold cannot be the domain of any nonconstant stable $\Phi$-harmonic map into any manifold, and $(\rm d)$ 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 also provide many examples of $\Phi$-SSU manifolds, and establish a link of $\Phi$-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.