Counter-example to a Kr\"oger type spectral inequality

Luc Pétiard

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arXiv · 1706.09189

Counter-example to a Kr\"oger type spectral inequality

Luc Pétiard

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Abstract

Given a Riemannian manifold, Weyl's law indicates how the spectrum of the Laplacian behaves asymptotically. Because of that result, there has been a growing interest in finding geometrical bounds compatible with this law. In the case of hypersurfaces, the isoperimetric ratio is a natural geometrical quantity, that allows to bound the spectrum from above. We investigate the problem and find an example of hypersurface where the eigenvalues are minorated by the isoperimetric ratio.

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Luc Pétiard. 2017-06-28. Counter-example to a Kr\"oger type spectral inequality. https://arxiv.org/abs/1706.09189

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