Some open problems and conjectures on submanifolds of finite type: recent development

Abstract

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

22 pages; to appeared in Tamkang J. Math. 45 (2014). arXiv admin note: substantial text overlap with arXiv:1307.6582

References in corpus (6)

• Recent developments of biharmonic conjecture and modified biharmonic conjectures
• Submanifolds with parallel mean curvature vector in Riemannian and indefinite space forms
• Sasaki manifolds, Kaehler cone manifolds and biharmonic submanifolds
• Biharmonic curves into quadrics
• Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold
• Biharmonic submanifolds into ellipsoids