On a certain class of finite type surfaces of revolution (Q1102560)

Let M be a complete connected surface of revolution in a Euclidean 3- space. The author proves that if each coordinate function of M is an eigenfunction of the Laplacian, then the surface is a catenoid, a sphere or a right circular cylinder. This result gives a partial solution to the following conjecture of the reviewer: Ordinary spheres are the only finite type compact surfaces in a Euclidean 3-space.