Advertisement

Geometriae Dedicata

, Volume 178, Issue 1, pp 1–13 | Cite as

Lagrangian immersions in the product of Lorentzian two manifolds

  • Nikos Georgiou
Original Paper

Abstract

For Lorentzian two manifolds \((\Sigma _1,g_1)\) and \((\Sigma _2,g_2)\) we consider the two product para-Kähler structures \((G^{\epsilon },J,\Omega ^{\epsilon })\) defined on the product four manifold \(\Sigma _1\times \Sigma _2\), with \(\epsilon =\pm 1\). We show that the metric \(G^{\epsilon }\) is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures \(\kappa _1,\kappa _2\) of \(g_1,g_2\), respectively, are both constants satisfying \(\kappa _1=-\epsilon \kappa _2\) (resp. \(\kappa _1=\epsilon \kappa _2\)). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product \(\gamma _1\times \gamma _2\subset \Sigma _1\times \Sigma _2\), where \(\gamma _1\) and \(\gamma _2\) are geodesics. We study Lagrangian surfaces in the product \(d{\mathbb S}^2\times d{\mathbb S}^2\) with parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and \(H\)-minimal surfaces.

Keywords

Lorentzian surfaces Para-Kaehler structure Minimal Lagrangian surfaces Surfaces with parallel mean curvarture vector Hamiltonian minimal surfaces 

Mathematics Subject Classification

Primary: 51M09 Secondary: 51M30 

Notes

Acknowledgments

The author would like to thank B. Guilfoyle and W. Klingenberg for their helpful and valuable suggestions and comments.

References

  1. 1.
    Alekseevsky, D., Guilfoyle, B., Klingenberg, W.: On the geometry of spaces of oriented geodesics. Ann. Global Anal. Geom. 40, 389–409 (2011)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Anciaux, H.: Minimal Submanifolds in Pseudo-Riemannian Geometry. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)MATHGoogle Scholar
  3. 3.
    Anciaux, H.: Space of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces. Trans. Am. Math. Soc. 366, 2699–2718 (2014)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Anciaux, H., Georgiou, N.: Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para- Kähler manifolds. Adv. Geom (in press)Google Scholar
  5. 5.
    Anciaux, H., Romon, P.: A canonical structure of the tangent bundle of a pseudo- or para- Kähler manifold. arxiv:1301.4638
  6. 6.
    Castro, I., Urbano, F.: Minimal Lagrangian surfaces in \({\mathbb{S}}^2\times {\mathbb{S}}^2\). Commun. Anal. Geom. 15, 217–248 (2007)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dong, Y.: On indefinite special Lagrangian submanifolds in indefinite complex Euclidean spaces. J. Geom. Phys. 59, 710–726 (2009)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Georgiou, N.: On minimal Lagrangian surfaces in the product of Riemannian two manifolds. Tôhoku Math. J (in press)Google Scholar
  9. 9.
    Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Harvey, R., Lawson, B.: Split special Lagrangian geometry. In: Dai, X., Rong, X. (eds.) Metric and Differential Geometry. The Jeff Cheeger Anniversary Volume, Progress in Mathematics, vol. 297, pp. 43–89. Springer, Berlin (2012)Google Scholar
  11. 11.
    Oh, Y.G.: Second variation and stabilities of minimal lagrangian submanifolds in Kähler manifolds. Invent. Math. 101, 501–519 (1990)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Oh, Y.G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Urbano, T., Urbano, F.: Surfaces with parallel mean curvature vector in \({\mathbb{S}}^2\times {\mathbb{S}}^2\) and \({\mathbb{H}}^2\times {\mathbb{H}}^2\). Trans. Am. Math. Soc. 364, 785–813 (2012)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Department of MathematicsFederal University of São CarlosSão CarlosBrazil

Personalised recommendations