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Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary

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Abstract

On a compact Riemannian manifold whose boundary is endowed with a Riemannian flow, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic 1-forms. The equality case gives rise to a particular geometry of the flow and of the boundary. Namely, we prove that the flow is a local product and the boundary is \(\eta \)-umbilical. This allows to characterize the quotient of \({\mathbb {R}}\times B'\) by some group \(\Gamma \) as the limiting manifold. Here \(B'\) denotes the unit closed ball. Finally, we deduce several rigidity results describing the product \({\mathbb {S}}^1\times {\mathbb {S}}^n\) as the boundary of a manifold.

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Acknowledgements

We would like to thank Nicolas Ginoux and Ken Richardson for many enlighting discussions and for pointing us a mistake in a previous version of the paper. 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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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences II, Lebanese University, P. O. Box 90656, Fanar-Matn, Lebanon

    Fida El Chami, Georges Habib & Ola Makhoul

  2. Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaizé, P. O. Box 72, Zouk Mikael, Lebanon

    Roger Nakad

Authors
  1. Fida El Chami
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  2. Georges Habib
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  3. Ola Makhoul
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  4. Roger Nakad
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Correspondence to Roger Nakad.

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El Chami, F., Habib, G., Makhoul, O. et al. Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary. Ricerche mat 67, 765–784 (2018). https://doi.org/10.1007/s11587-017-0339-7

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  • DOI: https://doi.org/10.1007/s11587-017-0339-7

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