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Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

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Abstract

In this paper we provide the second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian–Sasakian manifolds and relate the L-stability of L-minimal Legendrian submanifolds in a Sasakian manifold M to their L-stability in an associated Lorentzian–Sasakian structure on M.

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Notes

  1. Unlike Takahashi, in this paper we use the convention \(d\eta (X,Y) = X \eta (Y) - Y \eta (X) - \eta ([X,Y])\).

  2. To keep notation short, we later write \(\overline{\nabla }\) for \(f^*\overline{\nabla }\) and g for \(f^*g\).

  3. Beware that O’Neill uses the opposite convention than ours for Riemannian curvature.

  4. The formula on p. 609 of [3] differs by a sign as they define \(\varDelta = {{\mathrm{div}}}(\nabla \cdot )\).

References

  1. Alekseevsky, D.V., Medori, C., Tomassini, A.: Homogeneous para-Kählerian Einstein manifolds. Uspekhi Mat. Nauk 64(1) (385), 3–50 (2009)

  2. Anciaux, H.: Minimal Submanifolds in Pseudo-Riemannian Geometry. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, With a foreword by F. Urbano (2011)

  3. Anciaux, H., Georgiou, N.: Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds. Adv. Geom. 14(4), 587–612 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baum, H.: Twistor spinors on Lorentzian symmetric spaces. J. Geom. Phys. 34(3–4), 270–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, vol. 203, 2nd edn. Birkhäuser Boston, Inc., Boston (2010)

    Book  MATH  Google Scholar 

  6. Bohle, C.: Killing spinors on Lorentzian manifolds. J. Geom. Phys. 45(3–4), 285–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boyer, C., Galicki, K.: Sasakian Geometry. Oxford Science Publications, Oxford (2007)

    Book  MATH  Google Scholar 

  8. Boyer, C.P., Galicki, K., Matzeu, P.: On eta-Einstein Sasakian geometry. Commun. Math. Phys. 262(1), 177–208 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brunetti, L., Pastore, A.M.: On the classification of Lorentzian Sasaki space forms. Publ. Inst. Math. (Beograd) (N.S.) 94(108), 163–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Calamai, S., Petrecca, D.: On minimal Legendrian submanifolds of Sasaki–Einstein manifolds. Int. J. Math. 25(9), 1450083, 16 (2014)

  11. Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mt. J. Math. 26(1), 83–115 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iriyeh, H.: Hamiltonian minimal Lagrangian cones in \({\bf C}^m\). Tokyo J. Math. 28(1), 91–107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kajigaya, T.: Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds. Tohoku Math. J. (2) 65(4), 523–543 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Oh, Y.-G.: Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds. Invent. Math. 101(2), 501–519 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Oh, Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212(2), 175–192 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ohnita, Y.: On deformation of 3-dimensional certain minimal Legendrian submanifolds. In: Proceedings of the 13th International Workshop on Differential Geometry and Related Fields, vol. 13, , pp. 71–87. Natl. Inst. Math. Sci. (NIMS), Taejŏn (2009)

  17. Ohnita, Y.: Geometry of Certain Lagrangian Submanifolds in Hermitian symmetric spaces, Differential Geometry of Submanifolds and Its Related Topics. World Sci. Publ, Hackensack (2014)

    MATH  Google Scholar 

  18. O’Neill, B.: Semi-Riemannian Geometry, Pure and Applied Mathematics, vol. 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, With applications to relativity (1983)

  19. Ono, H.: Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds. Differ. Geom. Appl. 22(3), 327–340 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schäfer, L.: Lorentzian Legendrian mean curvature flow. Math. Z. 274(3–4), 1093–1111 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schoen, R., Wolfson, J.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 58(1), 1–86 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  23. Takahashi, T.: Sasakian manifold with pseudo-Riemannian metric. Tôhoku Math. J. (2) 21, 271–290 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tanno, S.: The topology of contact Riemannian manifolds. Ill. Math. J. 12, 700–717 (1968)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for his/her useful comments and remarks. They were supported by the Research Training Group 1463 “Analysis, Geometry and String Theory” of the DFG and the first author is supported as well by the GNSAGA of INdAM. They also would like to thank Fabio Podestà for his interest in their work.

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  1. Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hanover, Germany

    David Petrecca & Lars Schäfer

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  1. David Petrecca
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  2. Lars Schäfer
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Correspondence to David Petrecca.

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Communicated by A. Constantin.

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Petrecca, D., Schäfer, L. Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds. Monatsh Math 184, 273–289 (2017). https://doi.org/10.1007/s00605-017-1090-6

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