$n$-harmonicity, minimality, conformality and cohomology
Main Article Content
Abstract
By studying cohomology classes that are related with n-harmonic morphisms and F-harmonic maps, we augment and extend several results on F-harmonic maps, harmonic maps in [1, 3, 14], p-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and n-harmonic morphisms which are submersions. The results, for example Theorem 3.2 obtaine by utilizing the n-conservation law (2.6), are sharp.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
bibitem{A} M. Ara, {em Geometry of F-harmonic maps}. Kodai Math. J. {bf 22}(2) (1999), 243--263.
bibitem{Ba} P. Baird, {it Stress-energy tensors and the Lichnerowicz
Laplacian}, J. Geom. Phys. {bf 58}(10) (2008), 1329--1342.
bibitem{BE} P. Baird and J. Eells, {em A conservation law for harmonic maps}. Geometry Symposium, Utrecht 1980 (Utrecht, 1980), 1--25, Lecture Notes in Math. {bf 894} Springer, Berlin-New York, 1981.
bibitem{BG} P. Baird and S. Gudmundsson, {it $p$-harmonic maps and minimal submanifolds}. Math. Ann. {bf 294}(4) (1992), 611--624.
bibitem{BL} J. M. Burel and E. Loubeau, {em $p$-harmonic morphisms: the $1< p < 2$ case and a non-trivial example}. Contemp. Math. {bf 308} (2002), 21--37.
bibitem{c2005} B.-Y. Chen, {it Riemannian submersions, minimal immersions and cohomology class}.
Proc. Japan Acad. Ser. A Math. Sci. {bf 81}(10) (2005), 162--167.
bibitem{book11} B.-Y. Chen, {it Pseudo-Riemannian geometry, $delta$-invariants and applications}. World Scientific, Hackensack, NJ, 2011.
bibitem{book19} B.-Y. Chen, {it Geometry of Submanifolds}. Dover Publications, Mineola, NY, 2019. ISBN: 978-0-486-83278-4
bibitem{CW08} B.-Y. Chen and S. W. Wei, {it Geometry of submanifolds of warped product Riemannian
manifolds $Itimes_f S^{m-1}(k)$ from a p-harmonic viewpoint}. Bull. Transilv. Univ. Brac{s}ov Ser. III {bf 1(50)} (2008), 59--77.
bibitem{CW09} B.-Y. Chen and S. W. Wei, {it $p$-harmonic morphisms, cohomology classes and submersions}. Tamkang J. Math. {bf 40}(4) (2009), 377--382.
bibitem{CW23} B.-Y. Chen and S. W. Wei, {it $F$-harmonic morphisms, submersions, minimality and cohomology classes}. Preprint.
bibitem{DW} Y. X. Dong and S. W. Wei, {it On vanishing theorems for vector bundle valued $p$-forms and their applications}. Comm. Math. Phys. {bf 304} (2011), 329--368.
bibitem{EL} J. Eells and L. Lemaire, {it A report on harmonic maps}. Bull. London Math. Soc. {bf 10} (1978), 1--68.
bibitem{EL2} J. Eells and L. Lemaire, {it Selected topics in harmonic maps}. CBMS Regional Conf. Ser. in
Math., no. 50, Amer. Math. Soc., Providence, R.I., 1983.
bibitem{H} W.,V.,D. Hodge, {it The theory and application of harmonic intergrals.} Cambridge University
Press, New York, NY, 1941 (2nd ed. 1952)
bibitem{LSC} M. Lu, X.,W. Shen and K.,R. Cai, {it Liouville type Theorem for $p-$forms valued on vector bundle} (Chinese), J. of Hangzhou Normal Univ. (Natural Sci.) (2008), no. 7, 96--100
bibitem{OW} Y. L. Ou and S. W. Wei, {em A classification and some constructions of $p$-harmonic morphisms}, Beitr"age Algebra Geom. {bf 45}(2) (2004), 637--647.
bibitem{T} H. Takeuchi, {em Some conformal properties of $p$-harmonic maps and a regularity for sphere-valued $p$-harmonic maps}. J. Math. Soc. Japan {bf 46}(2) (1994), 217--234.
bibitem{W2} S. W. Wei, {it On topological vanishing theorems and the stability of Yang-Mills fields}. Indiana Univ. Math. J. {bf 33}(4) (1984), 511--529.
bibitem{S.W.Wei 1} S. W. Wei, emph{Representing Homotopy Groups and Spaces of Maps by p-harmonic maps}. Indiana Univ. Math. J. {bf 47}(2) (1998), 625--670.
bibitem{W9} S. W. Wei, {it The unity of $p$-harmonic geometry}. Recent developments in geometry and analysis, 439--483, Adv. Lect. Math. (ALM) {bf 23}, Int. Press, Somerville, MA, 2012.
bibitem{WLW} S.W. Wei, J.F. Li, and L. Wu, {em
Generalizations of the Uniformization Theorem and Bochner's Method
in $p$-Harmonic Geometry}, Commun. Math. Anal. Conf. 01, (2008) 46--68.