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

TL;DR

This paper investigates the spectral properties of Euclidean hypersurfaces with nearly extremal extrinsic radius or first eigenvalue, revealing their spectrum closely resembles that of extremal spheres and exploring how additional bounds influence their geometry and topology.

Contribution

It establishes asymptotic spectral similarities between nearly extremal hypersurfaces and spheres, and analyzes the impact of supplementary bounds on their geometric and topological features.

Findings

Spectra of almost extremal hypersurfaces approximate that of extremal spheres.

Additional bounds on $v_M\\|\\B\\\|_\\alpha^n$ influence spectral and topological properties.

The study provides groundwork for further geometric shape analysis in a subsequent paper.

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.