On classification of some surfaces of revolution of finite type

Abstract

Abstract. In this article, we study the following problem of [5] :Classify allfinite type surfaces in a Euclidean 3-space E3. A sur-face M in a Euclidean 3-space is said to be of finite type if eachof its coordinate functions is a finite sum of eigenfunctions of theLaplacian operator on M with respect to the induced metric (cf.[1, 2]). Minimal surfaces are the simplest examples of surfaces offinite type, in fact, minimal surfaces are of 1-type. The spheres,minimal surfaces and circular cylinders are the only known exam-ples of surfaces of finite type in Es and it seems to be the onlyfinite type surfaces in Ez (cf. [5]). The firstauthor conjectured in[2] that spheres are the only compact finite type surfaces in Es.Since then, it was proved step by step and separately that finitetype tubes, finite type ruled surfaces, finitetype quadrics and finitetype cones are surfaces of the only known examples (cf. [2, 6, 7,10].) Our next natural target for this classification problem is theclass of surfaces of revolution. However, this case seems to bemuch difficultthan the other cases mentioned above. We thereforeinvestigate this classification problem for this class and obtain classi-fication theorems for surfaces of revolution which are either ofrational or of polynomial kinds (cf. §1 for the definitions). Asconsequence, further supports for the conjecture cited above areobtained.