Spectrum of hypersurfaces with small extrinsic radius or large λ₁ in euclidean spaces

الأصل · EN

In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or λ₁ 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 vₘαⁿ and show that their spectral and topological properties depends on the position of α with respect to the critical value M. The study of the metric shape of these extremal hypersurfaces will be done in AG1, using estimates of the present paper.