Anisotropic isoparametric hypersurfaces in Euclidean spaces

TL;DR

This paper classifies complete anisotropic isoparametric hypersurfaces in Euclidean spaces, revealing their local and global properties and drawing parallels with classical isoparametric hypersurfaces.

Contribution

It provides a classification of anisotropic isoparametric hypersurfaces in Euclidean spaces, extending classical results to the anisotropic setting.

Findings

Complete classification of anisotropic isoparametric hypersurfaces.

Existence of local anisotropic isoparametric surfaces with both local and global properties.

Connection to proper Dupin hypersurfaces.

Abstract

In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces.