Abstract
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space \(\mathbb C P^N\) instead of submanifolds of \(\mathbb R ^N\) and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.
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Notes
We say that \(M\) is convex outside of \(D\), if after a perturbation of \(M\) which is the identity outside of \(D\) we get a convex compact hypersurface.
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Acknowledgments
This paper is a part of the author’s PhD thesis under the supervision of Professors Bruno Colbois (Neuchâtel University), Ahmad El Soufi (François Rabelais University), and Alireza Ranjbar-Motlagh (Sharif University of Technology) and she acknowledges their support and encouragement. The author wishes also to express her thanks to Bruno Colbois and Ahmad El Soufi for suggesting the problem and for many helpful discussions. She is also grateful to Mehrdad Shahshahani and the referee for helpful comments.
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The author has benefitted from the support of boursier du gouvernement Français during her stay in Tours.
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Hassannezhad, A. Eigenvalues of the Laplacian and extrinsic geometry. Ann Glob Anal Geom 44, 517–527 (2013). https://doi.org/10.1007/s10455-013-9379-8
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DOI: https://doi.org/10.1007/s10455-013-9379-8