Advertisement

Journal of Geometry

, 109:7 | Cite as

Equivariant biharmonic maps between manifolds with metrics of signature

  • Yuan-Jen Chiang
Article

Abstract

We obtain the reduction theorem of equivariant biharmonic maps between warped product manifolds, and apply it to punctured Euclidean spaces, spheres and hyperboloids. We also investigate equivariant biharmonic maps from equivariant manifolds with metrics \((p, \, q)\)-signature into equivariant manifolds with \((p', \, q')\)-signature, prove the reduction theorem, and provide a few examples.

Keywords

Biharmonic map equivariant (p, q)-manifold equivariant biharmonic map 

Mathematics Subject Classification

58E20 58A05 34A26 

References

  1. 1.
    Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds. J. Geom. Phys. 57(2), 449–466 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertola, M., Gouthier, D.: Lie triple systems and warped products. Rend. Math. Appl. 21(7), 275–293 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chiang, Y.J.: Developments of Harmonic Maps, Wave Maps and Yang–Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang–Mills Fields. Frontiers in Mathematics, p. xxi+399. Birkhäuser, Springer, Basel (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chiang, Y.J.: f-Biharmonic maps between Riemannian manifolds. J. Geom. Symmetry Phys. 27, 45–58 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chiang, Y.J., Ratto, A.: Paying tribute to James Eells and Joseph H. Sampson: in commemoration of the 50th anniversary of their pioneering work on harmonic maps. Not. Am. Math. Soc. 62(4), 388–393 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chiang, Y.J., Sun, H.A.: 2-Harmonic totally real submanifolds in a complex projective space. Bull. Inst. Math. Acad. Sin. 27(2), 99–107 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chiang, Y.J., Sun, H.A.: Biharmonic maps on V-manifolds. Int. J. Math. Math. Sci. 27(8), 477–484 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chiang, Y.J., Wolak, R.: Transversally biharmonic maps between foliated Riemannian manifolds. Int. J. Math. 19(8), 981–996 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chiang, Y.J., Wolak, R.: Transversally f-harmonic and transversally f-biharmonic maps. JP J. Geom. Topol. 13(1), 93–117 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chiang, Y.J., Wolak, R.: Remarks of transversally f-biharmonic maps. Proc. Balkan Soc. Geom. DGDS 21, 38–49 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coron, J.M., Goldberg, S.G.: Explosion en terms fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris I(308), 339–344 (1989)Google Scholar
  12. 12.
    Ding, W.Y.: Blow-up of solutions of heat flow for harmonic maps. Adv. Math. 19, 80–92 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ding, W.Y.: Symmetric harmonic maps between spheres. Commun. Math. Phys. 118, 641–649 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eells, J., Ratto, A.: Harmonic Maps and Minimal Immersions with Symmetries. Methods of Ordinary Differential Equations Applied to Elliptic Variational Problems. Annals of Mathematics Studies, vol. 130. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  16. 16.
    Hornung, P., Moser, R.: Existence of equivariant biharmonic maps. Int. Math. Res. Not. IMRN 2016(8), 2397–2422 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jiang, G.Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. A 7(4), 389–402 (1986). (Chinese)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jiang, G.Y.: 2-Harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math. A 7(2), 130–144 (1986). (Chinese)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Montaldo, S., Ratto, A.: A general approach to equivariant biharmonic maps. Mediterr. J. Math. 10(2), 1127–1139 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Loubeau, E., Oniciuc, C.: On the biharmonic and harmonic indices of the Hopf map. Trans. Am. Math. Soc. 359(11), 5239–5256 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic, New York (1983)zbMATHGoogle Scholar
  22. 22.
    Pettinati, V., Ratto, A.: Existence and non-existence results for harmonic maps between spheres. Ann. Sc. Norm. Sup. Pisa IV 17, 273–282 (1990)zbMATHGoogle Scholar
  23. 23.
    Ratto, A.: Equivariant harmonic maps between manifolds with metrics of (p, q)-signature. Ann. L’I. H. P. Sec. C 6, 503–524 (1989)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Smith, R.T.: The second variation formula for harmonic mappings. Proc. Am. Math. Soc. 47(1), 364–385 (1975)MathSciNetGoogle Scholar
  25. 25.
    Smith, R.T.: Harmonic maps of spheres. Am. J. Math. 97, 229–236 (1975)CrossRefGoogle Scholar
  26. 26.
    Urakawa, H.: Equivariant harmonic maps between compact manifolds of cohomology one. Mich. Math. J. 40, 27–51 (1993)CrossRefzbMATHGoogle Scholar
  27. 27.
    Xin, Y.L.: Geometry of Harmonic Maps. Progress in Nonlinear Differential Equations and Their Applications, vol. 23. Birkhäuser, Boston (1996)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

Personalised recommendations