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

The Reilly and Hasanis–Koutroufiotis inequalities give sharp bounds on λ1λ1 and on the extrinsic radius of Euclidean hypersurfaces in terms of the L2L2 norm of their mean curvature. The equality case of these inequalities characterizes the Euclidean spheres. In this paper, we study the spectral properties of the almost extremal hypersurfaces. We prove that the spectrum of the limit sphere asymptotically appears in the spectrum of almost extremal hypersurfaces for these inequalities. We also construct some examples of extremizing sequences that prove that the limit spectrum can be essentially any closed subset of R+R+ that contains the spectrum of the limit sphere. We also provide natural sharp condition to recover exactly the spectrum of the unit sphere.