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On classification of some surfaces of revolution of finite type. (English) Zbl 0795.53003
One of the most interesting problems in the theory of submanifolds of finite type is the classification of all surfaces of finite type in Euclidean 3-space. Up to today, the only known examples are minimal surfaces, ordinary spheres and circular cylinders. It is conjectured by the first author of the present paper that the only compact surfaces of finite type in Euclidean 3-space are the ordinary spheres. Quadrics and ruled surface of finite type have been classified by the first author, the reviewer, L. Verstraelen and L. Vrancken. The next classical family of surfaces to be investigated are the surfaces of revolution. Let \((f(u)\cos v,f(u)\sin v, g(u))\) be a parametrization on a surface of revolution \(M\) of finite type. The results in this paper are the following: (1) if both \(f\) and \(g\) are polynomial, then \(M\) is a plane or a circular cylinder; (2) if \(f(u)=u\) and \(g'(u)^ 2\) is a rational function \(Q(u)/R(u)\), then either \(M\) is a plane or a catenoid or else \(\det Q=\deg R=\deg (Q+R)+2\); (3) if \(f(u)=u\) and \(g\) is a rational function, then \(M\) is a plane.
Reviewer: F.Dillen (Leuven)

53A05 Surfaces in Euclidean and related spaces
53B25 Local submanifolds
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