Abstract
In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Bérard-Bergery and J. P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois Journal of Mathematics 26 (1982), 181–200.
M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics 194, Springer, New York, 1971.
D. D. Bleecker, Gauge Theory and Variational Principles, Global Analysis, Pure and Applied, No. 1, Addison-Wesley, Reading, MA, 1981.
V. Borrelli and O. Gil-Medrano, A critical radius for unit Hopf vector fields on spheres, Mathematische Annalen 334 (2006), 731–751
F. Brito, Total Bending of flows with mean curvature correction, Differential Geometry and its Applications 12 (2000), 157–163.
O. Gil-Medrano, Relationship between volume and energy of vector fields, Differential Geometry and its Applications 15 (2001), 137–152.
O. Gil-Medrano and A. Hurtado, Spacelike energy of timelike unit vector fields on a Lorentzian manifold, Journal of Geometry and Physics 51 (2004), 82–100.
O. Gil-Medrano and A. Hurtado, Volume, energy and generalized energy of unit vector fields on Berger’s spheres. Stability of Hopf vector fields, Proceedings of the Royal Society of Edinburgh Section A 135A (2005), 789–813.
O. Gil-Medrano and E. Llinares-Fuster, Second variation of Volume and Energy of vector fields. Stability of Hopf vector fields, Mathematische Annalen 320 (2001), 531–545.
O. Gil-Medrano and E. Llinares-Fuster, Minimal unit vector fields, The Tôhoku Mathematical Journal 54 (2002), 71–84.
H. Gluck and W. Ziller, On the volume of a unit vector field on the three-sphere, Commentarii Mathematici Helvetici 61 (1986), 177–192.
A. Hurtado, Volume, energy and spacelike energy of vector fields on Lorentzian manifolds, Proceedings of the II Meeting of Lorentzian Geometry Murcia, November 2003, Publ. de la RSME 8 (2004), 89–94.
A. Hurtado, Generalized energy of Hopf vector fields on Berger’s 3-spheres, Proc. Conf. Prague, August 30–September 3, 2004, Charles University, Prague (Czech Republic), 2005, pp. 69–77.
E. Loubeau and C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Transactions of the American Mathematical Society 359 (2007), 5239–5256.
H. J. Rivertz and P. Tomter, Stability of geodesic spheres, in Geometry and Topology of Submanifolds, VII (Brussels, 1995/Nordjordeid, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 320–324.
S. Tanno, The first eigenvalue of the Laplacian on spheres, The Tôhoku Mathematical Journal 31 (1979), 179–185.
G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Mathematische Annalen 303 (1995), 325–344.
C. M. Wood, On the energy of a unit vector field, Geometriae Dedicata 64 (1997), 319–330.
C. M. Wood, The energy of Hopf vector fields, Manuscripta Mathematica 101 (2000), 71–88.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by DGI (Spain) and FEDER Project MTM 2004-06015-C02-01 and by Generalitat Valenciana Grant ACOMP06/166.
Rights and permissions
About this article
Cite this article
Hurtado, A. Instability of Hopf vector fields on Lorentzian Berger spheres. Isr. J. Math. 177, 103–124 (2010). https://doi.org/10.1007/s11856-010-0039-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-010-0039-4