Surfaces of coordinate finite II-type
Hassan Al-Zoubi, Mutaz Al-Sabbagh, Tareq Hamadneh
TL;DR
The paper investigates surfaces of revolution in $E^{3}$ with nonvanishing Gauss curvature for which the position vector $\boldsymbol{x}$ satisfies $\Delta^{II}\boldsymbol{x}=A\boldsymbol{x}$ for some $3\times3$ matrix $A$, i.e., they are of coordinate finite type with respect to the second fundamental form. By parameterizing revolution surfaces as $\boldsymbol{x}(u,v)=(p(u)\cos v, p(u)\sin v, q(u))$ and computing $\Delta^{II}$ in these coordinates, the problem is reduced to a system that constrains $A$, followed by a comprehensive case analysis of eigenvalues. The main result is that the only such surfaces are the sphere and the catenoid; other eigenvalue configurations yield contradictions. This work extends the finite $II$-type framework to a classification result for revolution surfaces, linking a second fundamental form Laplacian condition to classical geometric objects like spheres and minimal surfaces.
Abstract
In this article, we study the class of surfaces of revolution in the 3-dimensional Euclidean space $E^{3}$ with nonvanishing Gauss curvature whose position vector $\boldsymbol{x}$ satisfies the condition $Δ^{II}\boldsymbol{x}=A\boldsymbol{x}$, where $A$ is a square matrix of order 3 and $Δ^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.