On a family of surfaces of revolution of finite Chen-type (Q1267472)

The only known finite type surfaces in \( E ^3\) are portions of spheres, minimal surfaces and of circular cylinders. The following conjecture was formulated by B.-Y. Chen: The only compact surfaces of finite type in \( E ^3\) are the spheres. The conjecture is still open. Several authors have confirmed the conjecture in special cases. In the paper under review, the authors confirm Chen's conjecture for compact surfaces \(M\) of revolution under the assumption that there are at least two ellipses contained in \(M\) through each point of \(M\).