Rotational hypersurfaces family satisfying Ln−3G=AG in the n-dimensional Euclidean space

Abstract

In this paper, we investigate rotational hypersurfaces family in n-dimensional Euclidean space $E^n$. Our focus is on studying the Gauss map $G$ of this family with respect to the operator $L_k$, which acts on functions defined on the hypersurfaces. The operator $L_k$ can be viewed as a modified Laplacian and is known by various names, including the Cheng–Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\ge 3$. By applying the operator $L_{n-3}$ to the Gauss map $G$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $A$, and the Gauss map $G$ through the equation $L_{n-3}G=AG$.