Constant curved minimal CR 3-spheres in CPⁿ

Abstract

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 <p :is uniquely determined by the induced metric, and <p can be constructed from its directrix <p Q : S 2 -> 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).

The Minimal With Constant Sectional Curvature In

“…From Theorem 4.3, we conclude that a minimal immersion of CR type is equivariant and, when k = 1, from (4.9) we know that 0 ≤ c ≤ 1/3. Together with Theorem 4.2, we also get the following rigidity result [7]. Theorem 4.4.…”

“…So, one may ask whether the minimal S 3 with constant sectional curvature isometrically immersed into CP n also has rigidity. Li [6] and Li and Huang [7] proved that this is true for the minimal S 3 of CR type with constant sectional curvature in CP n .…”

“…In this paper, besides the CR-type examples in [6], we give some new examples by the unitary representation of SU (2), which are neither weakly Lagrangian nor of CR type. By using the method of moving frames in [4], we get the adapted frame of the minimal S 3 of CR type with constant sectional curvature in CP n , and also give a new proof of the rigidity theorem in [7].…”

Equivariant CR minimal immersions from $$S^3$$S3 into $$\mathbb CP^n$$CPn

“…To extend the research of [2], and as a counterpart of Mashimo [17] studying the immersions from S 3 into S n , a variety of immersions of S 3 into CP n have been considered. For related references we refer to [3,7,8,12,13,14,15,16], among which the paper of the third author [12] is of fundamental importance for us.…”

“…Moreover, up to an isometry of S 3 and a holomorphic isometry of CP n , the immersion ϕ is exactly the immersion defined in Example 4.4 of [12]. In [13], Li and Huang made an interesting advance by showing that any minimal CR 3-sphere in CP n is actually equivariant if the induced metric of S 3 has constant sectional curvature.…”