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

Abstract

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].

Global Properties of Biconservative Surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$

“…Thus,f satisfies a polynomial equation with constant coefficients, sof has to be a constant and then, f is a constant, i.e., grad f = 0 on U (in fact, f has to be zero). Therefore, we have a contradiction (see [6,8] for c = 0 and [3,4], for c = ±1).…”

On Slant Curves with pseudo-Hermitian $C$-parallel Mean Curvature Vector Fields

“…In [7], Chen defined biharmonic submanifold as a Riemannian submanifold with vanishing Laplacian of mean curvature vector field ∆H. Curves in a Euclidean space satisfying the condition ∆ ⊥ H = λH were classified in [2], by Barros and Garay, where ∆ ⊥ denotes the Laplacian of the curve in the normal bundle and λ is a real valued function.…”

Biminimal properly immersed submanifolds in the Euclidean spaces

“…[2,3,4,6,7,9]). In particular, there are some complete affirmative answers if M is one of the following: (a) a curve [7], (b) a surface in E 3 [2], (c) a hypersurface in E 4 [6,9]. On the other hand, since there is no assumption of completeness for submanifolds in Conjecture 1, in a sense it is a problem in local differential geometry.…”