Cheng-Yau operator and Gauss map of surfaces of revolution

TL;DR

This paper classifies surfaces of revolution in Euclidean 3-space whose Gauss map satisfies a specific differential equation involving the Cheng-Yau operator, identifying only planes, cones, cylinders, and spheres as solutions.

Contribution

It provides a complete classification of surfaces of revolution with Gauss maps satisfying the Cheng-Yau operator equation, extending understanding of geometric properties of such surfaces.

Findings

Planes, cones, cylinders, and spheres satisfy the Cheng-Yau operator condition.

The classification theorem characterizes all surfaces of revolution with this property.

The Gauss map condition leads to a unique set of classical surfaces.

Abstract

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 $\square$ 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 $\square G=AG$ for some $3\times3$ matrix $A$ are the planes, right circular cones, circular cylinders and spheres.