Title: The geometry of $\Phi_{(3)}$-harmonic maps
Authors: Shuxiang Feng, Yingbo Han, Kaige Jiang and Shihshu Walter Wei
Categories: math.DG math-ph math.AP math.MP
Comments: 46 pages, to appear in Nonlinear Analysis (2023). arXiv admin note:
text overlap with arXiv:1911.05855
MSC-class: 58E20, 53C21, 53C25
\\
In this paper, we motivate and extend the study of harmonic maps or
$\Phi_{(1)}$-harmonic maps (cf [15], Remark 1.3 (iii)), $\Phi$-harmonic maps or
$\Phi_{(2)}$-harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric
properties of $\Phi_{(3)}$-harmonic maps by unified geometric analytic methods.
We define the notion of $\Phi_{(3)}$-harmonic maps and obtain the first
variation formula and the second variation formula of the $\Phi_{(3)}$-energy
functional $E_{\Phi_{(3)}}$. By using a stress-energy tensor, the
$\Phi_{(3)}$-conservation law, a monotonicity formula, and the asymptotic
assumption of maps at infinity, we prove Liouville type results for
$\Phi_{(3)}$-harmonic maps. We introduce the notion of
$\Phi_{(3)}$-Superstrongly Unstable ($\Phi_{(3)}$-SSU) manifold and provide
many interesting examples. By using an extrinsic average variational method in
the calculus of variations (cf. [51, 49]), we find $\Phi_{(3)}$-SSU manifold
and prove that for $i=1,2,3$, every compact $\Phi_{(i)}$-$\operatorname{SSU}$
manifold is $\Phi_{(i)}$-$\operatorname{SU}$, and hence is
$\Phi_{(i)}$-$\operatorname{U}$ (cf. Theorem 9.3). As consequences, we obtain
topological vanishing theorems and sphere theorems by employing a
$\Phi_{(3)}$-harmoic map as a catalyst. This is in contrast to the approaches
of utilizing a geodesic ([45]), minimal surface, stable rectifiable current
([34, 29, 50]), $p$-harmonic map (cf. [53]), etc., as catalysts. These
mysterious phenomena are analogs of harmonic maps or $\Phi_{(1)}$-harmonic
maps, $p$-harmonic maps, $\Phi_{S}$-harmonic maps, $\Phi_{S,p}$-harmonic maps,
$\Phi_{(2)}$-harmonic maps, etc., (cf. [21, 40, 42, 41, 12, 13]).
\\ ( https://arxiv.org/abs/2305.19503 , 30kb)