Surfaces of revolution satisfying $\triangle^{III}\boldsymbol{x}=A\boldsymbol{x}$

TL;DR

This paper classifies surfaces of revolution in 3D space that satisfy a specific differential equation involving the third fundamental form, showing they are either catenoids or parts of spheres.

Contribution

It provides a complete classification of revolution surfaces satisfying the differential relation with the third fundamental form, identifying them as catenoids or spherical segments.

Findings

Surfaces of revolution satisfying the relation are catenoids or spherical segments.

The classification is complete for surfaces of coordinate finite type with respect to the third fundamental form.

Abstract

We consider surfaces of revolution in the three-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.