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

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


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.


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 



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.


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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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