Metric shape of hypersurfaces with small extrinsic radius or large ${\lambda }_1$

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.