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A report on submanifolds of finite type. (English) Zbl 0867.53001

Author’s abstract: “The study of submanifolds of finite type began in the late 1970’s through the author’s attempts to find the best possible estimate of the total mean curvature of a compact submanifold of a Euclidean space and to find a notion of “degree” for submanifolds of a Euclidean space. Similar to minimal submanifolds, submanifolds of finite type are characterized by a variational minimal principle in a natural way. The first results on this subject have been collected in the author’s books [Total mean curvature and submanifolds of finite type, World Scientific, Singapore (1984; Zbl 0537.53049); Finite type submanifolds and generalizations, Universita degli Studi di Roma “La Sapienza”, Roma (1985; Zbl 0586.53023)] more than a decade ago. Since that time, the subject has had a rapid development. The main purpose of this survey article is to report the progress made on this subject by various geometers during the past decade and also to present a picture of this subject as it is currently evolving. In addition, we provide a few new results in this subject. In the last section, we include some open problems and conjectures in this area of research.”
The new results concern mainly constructions and characterizations of finite type submanifolds in symmetric spaces. The bibliography of this comprehensive survey contains 348 entries.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
49Q20 Variational problems in a geometric measure-theoretic setting
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
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