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

## TL;DR

This paper presents a counter-example to a spectral inequality related to the isoperimetric ratio on hypersurfaces, challenging previous assumptions about geometric bounds on Laplacian eigenvalues.

## Contribution

It provides the first known example where the eigenvalues are bounded below by the isoperimetric ratio, contradicting expected inequalities.

## Key findings

- Counter-example to a Kröger type spectral inequality
- Eigenvalues are minorated by the isoperimetric ratio in the example
- Challenges existing conjectures on geometric bounds for spectra

## 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.