# Spectrum of hypersurfaces with small extrinsic radius or large $\lambda_1$ in euclidean spaces

1 Géométrie et Analyse
JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or $\lambda_1$ have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on $v_M\|\B\|_\alpha^n$ and show that their spectral and topological properties depends on the position of $\alpha$ with respect to the critical value $\dim M$. The study of the metric shape of these extremal hypersurfaces will be done in \cite{AG1}, using estimates of the present paper.
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Journal articles

https://hal.archives-ouvertes.fr/hal-00743727
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Erwann Aubry, Jean-Francois Grosjean. Spectrum of hypersurfaces with small extrinsic radius or large $\lambda_1$ in euclidean spaces. Journal of Functional Analysis, Elsevier, 2016, 271 (5), pp.1213-1242. ⟨10.1016/j.jfa.2016.06.011⟩. ⟨hal-00743727v2⟩

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