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.
Similar content being viewed by others
References
Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds. J. Geom. Phys. 57(2), 449–466 (2007)
Bertola, M., Gouthier, D.: Lie triple systems and warped products. Rend. Math. Appl. 21(7), 275–293 (2001)
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)
Chiang, Y.J.: f-Biharmonic maps between Riemannian manifolds. J. Geom. Symmetry Phys. 27, 45–58 (2012)
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)
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)
Chiang, Y.J., Sun, H.A.: Biharmonic maps on V-manifolds. Int. J. Math. Math. Sci. 27(8), 477–484 (2001)
Chiang, Y.J., Wolak, R.: Transversally biharmonic maps between foliated Riemannian manifolds. Int. J. Math. 19(8), 981–996 (2008)
Chiang, Y.J., Wolak, R.: Transversally f-harmonic and transversally f-biharmonic maps. JP J. Geom. Topol. 13(1), 93–117 (2013)
Chiang, Y.J., Wolak, R.: Remarks of transversally f-biharmonic maps. Proc. Balkan Soc. Geom. DGDS 21, 38–49 (2014)
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)
Ding, W.Y.: Blow-up of solutions of heat flow for harmonic maps. Adv. Math. 19, 80–92 (1990)
Ding, W.Y.: Symmetric harmonic maps between spheres. Commun. Math. Phys. 118, 641–649 (1988)
Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
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)
Hornung, P., Moser, R.: Existence of equivariant biharmonic maps. Int. Math. Res. Not. IMRN 2016(8), 2397–2422 (2016)
Jiang, G.Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. A 7(4), 389–402 (1986). (Chinese)
Jiang, G.Y.: 2-Harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math. A 7(2), 130–144 (1986). (Chinese)
Montaldo, S., Ratto, A.: A general approach to equivariant biharmonic maps. Mediterr. J. Math. 10(2), 1127–1139 (2013)
Loubeau, E., Oniciuc, C.: On the biharmonic and harmonic indices of the Hopf map. Trans. Am. Math. Soc. 359(11), 5239–5256 (2007)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic, New York (1983)
Pettinati, V., Ratto, A.: Existence and non-existence results for harmonic maps between spheres. Ann. Sc. Norm. Sup. Pisa IV 17, 273–282 (1990)
Ratto, A.: Equivariant harmonic maps between manifolds with metrics of (p, q)-signature. Ann. L’I. H. P. Sec. C 6, 503–524 (1989)
Smith, R.T.: The second variation formula for harmonic mappings. Proc. Am. Math. Soc. 47(1), 364–385 (1975)
Smith, R.T.: Harmonic maps of spheres. Am. J. Math. 97, 229–236 (1975)
Urakawa, H.: Equivariant harmonic maps between compact manifolds of cohomology one. Mich. Math. J. 40, 27–51 (1993)
Xin, Y.L.: Geometry of Harmonic Maps. Progress in Nonlinear Differential Equations and Their Applications, vol. 23. Birkhäuser, Boston (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chiang, YJ. Equivariant biharmonic maps between manifolds with metrics of signature. J. Geom. 109, 7 (2018). https://doi.org/10.1007/s00022-018-0408-4
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00022-018-0408-4