Various kinds of minimal 3-spheres with constant sectional curvature in the complex projective space CP n are studied.

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in H n+1 1 with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H n+1 1 or S n+1 1 can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces.

In this paper, we refine some results of [arXiv: 0808.1185v1]. As an application, let M be a complete [Formula: see text]-stable minimal hypersurface in an (n + 1)-dimensional Euclidean space ℝn+1 with n ≥ 3, we prove that if M has bounded norm of the second fundamental form, then M must have only one end. Moreover, we also prove that if M has finite total curvature, then M is a hyperplane.

In this paper we prove that minimal 3-spheres of CR type with constant sectional curvature c in the complex projective space CP" are all equivariant and therefore the immersion is rigid. The curvature c of the sphere should be c = \/{m 2 -1) for some integer m > 2, and the full dimension is n = 2m 2 -3. An explicit analytic expression for such an immersion is given.2000 Mathematics subject classification: primary 53C42; secondary 53C55. Keywords and phrases: minimal, constant curvature, CR-submanifold, complex projective space. PreliminaryIn In this paper, we assume that N is the complex projective space CP" with constant holomorphic sectional curvature 4.The minimal surface theory in CP" has made a great progress over the past thirty years. For constant curved minimal 2-spheres in CP", the immersion

CP" by using arithmetical procedure [2].Up to now merely a few examples have been known for higher dimensional minimal submanifolds in CP". There are some examples of holomorphic submanifolds and Lagrangian minimal submanifolds [3,4,6]. In [5] we studied equivariant minimal 3-spheres with constant (sectional) curvature c immersed in CP". Here the terminology Project supported by the NSFC (10261006), the NSFJP (0211005) and the FANEDD (200217).

No abstract