Anisotropic isoparametric hypersurfaces in Euclidean spaces

In this paper, by Nomizu's method and some technical treatment of the asymmetry of the F-Weingarten operator, we obtain a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is a generalization of 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 in general anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces, which differs from classical isoparametric hypersurfaces.