Cheng-Yau operator and Gauss map of surfaces of revolution

We study the Gauss map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}}^3$ with respect to the so called Cheng-Yau operator $\mathrm{\square }$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map $G$ satisfying $\mathrm{\square }G=AG$ for some $3\times 3$ matrix $A$ are the planes, right circular cones, circular cylinders and spheres.