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Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary, Part II

  • Fida El Chami
  • Georges Habib
  • Ola Makhoul
  • Roger Nakad
Article
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Abstract

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in El Chami et al. (Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, 2015) a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when \(p>1\). We then characterize the limiting case by showing that the manifold N is isometric to Open image in new window for some group \(\Gamma \) where \(B'\) denotes the unit closed ball. As a consequence, we describe the Riemannian product \({\mathbb {S}}^1\times {\mathbb {S}}^n\) as the boundary of a manifold.

Keywords

Riemannian flow manifolds with boundary basic Laplacian eigenvalue second fundamental form O’Neill tensor basic Killing forms rigidity results 

Mathematics Subject Classification

53C12 53C24 58J50 58J32 

Notes

Acknowledgements

The first two named authors were supported by a fund from the Lebanese University. The second named author would like to thank the Alexander von Humboldt Foundation for its support.

References

  1. 1.
    Belishev, M., Sharafutdinov, V.: Dirichlet to Neumann operator on differential forms. Bull. Sci. Math. 132, 128–145 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carrière, Y.: Flots riemanniens, structure transverse des feuilletages. Toulouse, Astérique 116, 31–52 (1984)Google Scholar
  3. 3.
    El Chami, F., Habib, G., Ginoux, N., Nakad, R.: Rigidity results for spin manifolds with foliated boundary. J. Geom. 107, 533–555 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    El Chami, F., Habib, G., Makhoul, O., Nakad, R. Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary (2015). (To appear in Ricerche di Matematica) Google Scholar
  5. 5.
    El Kacimi, A., Gmira, B.: Stabilité du caractère kählérien transverse. Isr. J. Math. 101, 323–347 (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    El Kacimi, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compositio Mathematica 73, 57–106 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gallot, S., Meyer, D.: Sur la première valeur propre du \(p\)-spectre pour les variétés à opérateur de courbure positif. C.R. Acad. Sci Paris 320, 1331–1335 (1995)zbMATHGoogle Scholar
  8. 8.
    Jung, S.D., Richardson, K.: Transversal conformal Killing forms and a Gallot-Meyer theorem for foliations. Math. Z. 270, 337–350 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, M.: A sharp comparison theorem for compact manifolds with mean convex boundary. J. Geom. Anal. 24, 1490–1496 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Raulot, S., Savo, A.: A Reilly formula and eigenvalue estimates for differential forms. J. Geom. Anal. 3, 620–640 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Richardson, K., Park, E.: The basic Laplacian of a Riemannian foliation. Am. J. Math. 118, 1249–1275 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Reinhart, B.: Foliated manifolds with bundle-like metrics. Ann. Math. 69, 119–132 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schwarz, G.: Hodge decomposition-A method for solving boundary value problems. Lecture notes in Mathematics, Springer (1995)Google Scholar
  15. 15.
    Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503–527 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tachibana, S., Yu, W.N.: On a Riemannian space admitting more than one Sasakian structures. Tohoku Math. J. 22, 536–540 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tondeur, Ph: Foliations on Riemannian manifolds. Springer, New York (1959)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Fida El Chami
    • 1
  • Georges Habib
    • 1
  • Ola Makhoul
    • 1
  • Roger Nakad
    • 2
  1. 1.Department of Mathematics, Faculty of Sciences IILebanese UniversityFanar-MatnLebanon
  2. 2.Department of Mathematics and Statistics, Faculty of Natural and Applied SciencesNotre Dame University-LouaizéZouk MikaelLebanon

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