On a certain class of finite type surfaces of revolution

Abstract

In 1983 B-Y Chen introduced the notion of Euclidean immersions of Finite Type. Essentially these are submanifolds whose immersion into E is constructed by making use of a finite number of £-valuated eigenfunctions of their Laplacian. In terms of Finite Type terminology, a well-known result of Takahashi [9], affirms that a connected Euclidean submanifold is of 1-type, if and only if it is either minimal in E or minimal in some hypersphere of E. So, one can consider the Finite Type submanifolds as a generalization of both minimal in E and minimal into spheres. Submanifolds of Finite Type closest in simplicity to the minimal ones are the 2-type spherical submanifolds (where spherical means into a sphere). Important results about 2-type spherical closed submanifolds have been obtained in [1], [2], [3], [7]. However relatively little is known about Finite Type submanifolds which are not contained into spheres. This is so in part, because the partial differential equations which appear in the study of such submanifolds, are very difficult of dealing with. The first attempt of liberating the Finite Type submanifolds of a spherical support, was made by B-Y Chen in [4], where he proves that a non-cylindrical tube is of Infinite Type. This result along with the fact that the only closed curve of Finite Type immersed in a Euclidean space E, is a circle [3], incites that author to do the following conjecture [5] : The only Finite Type closed plane surface into the 3-dimensional Euclidean space is the sphere. In this paper we want to determine the complete surfaces of revolution in E whose component functions are eigenfunctions of their Laplace operator. In the first step, we shall see that this kind of surfaces are at most of 2-tyρe. Then we shall reduce the geometric problem to a simpler ordinary differential equation system. By solving it, we obtain that a surface of this sort must be a catenoid, a right circular cylinder, or a sphere. In particular the only closed of them is the sphere, giving the first approximation to Chen's conjecture. Another consequence, is that they all have constant mean curvature. However, the mean curvature constancy is not a sufficient condition by a result of Kenmotsu, [8]. This fact is analogous to that which states that a 2-type mass-symmetric closed submanifold of S has constant mean curvature, [3], but if n—2>, then M must be flat, [2], [3], and this obviously does not occur in our case. Our result can