Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

Abstract : We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of $M$ with a $p$-plane in a generic position (transverse to $M$), or an invariant which measures the concentration of the volume of $M$ in $\R^{m+p}$. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when $p=1$), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for $m\ge 3$) the differential structure.
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Bruno Colbois, Emily Dryden, Ahmad El Soufi. Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds. Bulletin of the London Mathematical Society / The Bulletin of the London Mathematical Society, 2010, 42 (1), pp.96--108. ⟨10.1112/blms/bdp100⟩. ⟨hal-00420689⟩

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