Metric shape of hypersurfaces with small extrinsic radius or large $ \lambda_1$

TL;DR

This paper characterizes the limiting shapes of Euclidean hypersurfaces that have either a large first eigenvalue or a small extrinsic radius, depending on the boundedness of their curvature in specific $L^p$ norms.

Contribution

It establishes the Hausdorff limit-set of such hypersurfaces based on the $L^p$ curvature bounds, revealing critical behavior at a specific $p$ value.

Findings

Limit-set depends on $L^p$ curvature bounds

Critical behavior at $p = ext{dimension} - 1$

Characterization of hypersurfaces with extremal geometric properties

Abstract

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $\lambda_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for $p$ equal to the dimension minus 1.