Abstract
Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baird, P., Eells, J.: A conservation law for harmonic maps. Geometry Symposium, Utrecht 1980. Lecture Notes in Mathematics, vol. 894, pp. 1–25. Springer, Berlin (1981)
Baird P. and Kamissoko D. (2003). On constructing biharmonic maps and metrics. Ann. Glob. Anal. Geom. 23: 65–75
Baird P. and Wood J.C. (2003). Harmonic Morphisms between Riemannian Manifolds. Oxford Science Publications, Oxford
Balmuş A. (2004). Biharmonic properties and conformal changes. An. Stiint. Univ. Al.I.~Cuza Iasi Mat. (N.S.) 50: 361–372
Balmuş A., Montaldo S. and Oniciuc C. (2007). Biharmonic maps between warped product manifolds. J. Geom. Phys. 57: 449–466
Berger M. (1970). Quelques formules de variation pour une structure riemannienne. Ann. Sci. Ec. Norm. Sup. 3: 285–294
Besse A. (1987). Einstein Manifolds. Springer, Berlin
Caddeo R., Montaldo S. and Oniciuc C. (2001). Biharmonic submanifolds of \({\mathbb{S}}^3\). Int. J. Math. 12: 867–876
Caddeo R., Montaldo S. and Oniciuc C. (2002). Biharmonic submanifolds in spheres. Israel J. Math. 130: 109–123
Chen B.Y. (1973). Geometry of Submanifolds. Marcel Dekker, New York
Dillen F. (1991). Semi-parallel hypersurfaces of a real space form. Israel J. Math. 75: 193–202
Eells J. and Sampson J.H. (1964). Harmonic mappings of Riemannian manifolds. Am. J. Math. 86: 109–160
Eells J. and Lemaire L. (1978). A report on harmonic maps. Bull. Lond. Math. Soc. 10: 1–68
Hilbert D. (1924). Die Grundlagen der Physik. Math. Ann. 92: 1–32
Hu Z.J. (1995). Complete hypersurfaces with constant mean curvature and nonnegative sectional curvatures. Proc. Am. Math. Soc. 123: 2835–2840
Jiang G.Y. (1987). The conservation law for 2-harmonic maps between Riemannian manifolds. Acta Math. Sin. 30: 220–225
Montaldo S. and Oniciuc C. (2006). A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47(2): 1–22
Oniciuc C. (2002). Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al.I. Cuza Iasi Mat. (N.S.) 48: 237–248
Petersen P. (1998). Riemannian Geometry. Springer, New York
Ruh E. and Vilms J. (1970). The tension field of the Gauss map. Trans. Am. Math. Soc. 149: 569–573
Sakai T. (1996). On Riemannian manifolds admitting a function whose gradient is of constant norm. Kodai Math. J. 19: 39–51
Sanini A. (1983). Applicazioni tra varietà riemanniane con energia critica rispetto a deformazioni di metriche. Rend. Mat. 3: 53–63
Yau S.T. (1976). Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25: 659–670
Author information
Authors and Affiliations
Corresponding author
Additional information
In memoriam James Eells.
S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).
Rights and permissions
About this article
Cite this article
Loubeau, E., Montaldo, S. & Oniciuc, C. The stress-energy tensor for biharmonic maps. Math. Z. 259, 503–524 (2008). https://doi.org/10.1007/s00209-007-0236-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0236-y