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

Abstract

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.