On Laguerre Isotropic Hypersurfaces

TL;DR

This paper investigates Laguerre isotropic hypersurfaces in Euclidean space, establishing rigidity results, characterizing those that are also L-isoparametric, and deriving conditions on their curvature eigenvalues.

Contribution

It provides a rigidity theorem for L-isotropic hypersurfaces parametrized by lines of curvature and characterizes hypersurfaces that are both L-isotropic and L-isoparametric.

Findings

Rigidity theorem for L-isotropic hypersurfaces with curvature parametrization

Characterization of L-isotropic and L-isoparametric hypersurfaces with zero eigenvalue

Bound on the curvature-related function for hypersurfaces with positive eigenvalue

Abstract

We study Laguerre isotropic hypersurfaces in the Euclidean space, which are hypersurfaces whose Laguerre form is zero and the eigenvalues of the Laguerre tensor are constant and equal to $\lambda\geq 0$. We prove a rigidity theorem for the L-isotropic hypersurfaces parametrized by lines of curvature. Moreover, we study the hypersurfaces that are L-isotropic and L-isoparametric simultaneously and we show that for such a hypersurface $\lambda=0$. We obtain necessary conditions for the existence of L-isotropic hypersurfaces with $\lambda > 0$ and we prove that a certain function, determined by the radii of curvature of the hypersurface, is bounded above by ${1}/{2\lambda}$.