Rotational hypersurfaces family satisfying ${\mathbb{L}}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^n$

Abstract

In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${\mathbb{E}}^n$. Our focus is on studying the Gauss map $\mathcal{G}$ of this family with respect to the operator ${\mathbb{L}}_k$, which acts on functions defined on the hypersurfaces. The operator ${\mathbb{L}}_k$ 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 ${\mathbb{L}}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the equation ${\mathbb{L}}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$.

arXiv comment17 pages