Spectrum of hypersurfaces with small extrinsic radius or large ${\lambda }_1$ in euclidean spaces

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 $v_M\mathrm{\parallel }\text{B}{\mathrm{\parallel }}_{\alpha }^n$ 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.