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