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Some open problems and conjectures on submanifolds of finite type. (English) Zbl 0749.53037

Submanifolds of finite type were introduced by the present author more than ten years ago [Bull. Inst. Math., Acad. Sin. 7, 301-311 (1979; Zbl 0442.53050); 11, 309-328 (1983; Zbl 0498.53039)]. Since then this subject has been strongly developed, remarkable results being obtained by the author himself or in joint works. Many of the basic results were presented in his book [Total mean curvature and submanifolds of finite type (World Scientific 1984; Zbl 0537.53049)]. As one can see in the title, in the present paper are presented many problems and conjectures about this theory. For many of those problems there are known some partial results, but they are still open and seem to be quite difficult to solve. We can quote some of them: 1) The only compact finite type surfaces in ${E}^{3}$ are the spheres. 2) Classify finite type hypersurfaces of a hypersphere in ${E}^{n+2}$. 3) Minimal surfaces, standard 2-spheres and products of plane circles are the only finite type surfaces in ${S}^{3}$ (imbedded standardly in ${E}^{4}$). 4) Is every $n$- dimensional non-null 2-type submanifold of ${E}^{n+2}$ with constant mean curvature spherical? 5) The only biharmonic submanifolds in Euclidean spaces are the minimal ones.

In my opinion, this field of submanifolds of finite type is very interesting and with nice perspectives.

##### MSC:
 53C40 Global submanifolds (differential geometry) 53-02 Research monographs (differential geometry) 53B25 Local submanifolds