MINIMAL $S^3$ WITH CONSTANT CURVATURE IN $\mathbb{C}^n$

Abstract

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

Constant curved minimal CR 3-spheres in CPn

“…Up to an isometry of S 3 we may consider the metric ds 2 as a bi-invariant metric on SU(2). Two maps <p, \jr : S 3 -> CP" are said to be equivalent if there is a holomorphic isometric A : -> C P " C P " such that \jf = A o (p. We have the following results from [5]. …”

“…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).…”

“…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 2 Zhen-Qi Li and An-Min Huang [2] 'equivariant' means that the immersion <p : S 3 ->• C P " is compatible with the Lie group structure on S 3 = SU (2), that is, there exists a homomorphism E : S 3 -> U(/i + 1) of Lie group such that (p = A o 7r 2 O E, where…”

“…In [5], we provided two examples of minimal immersions from S 3 into CP". One of these examples is below.…”

“…The bi-invariant metric ds 2 defines an inner product on su(2) in a natural way, and induces the inner product on su(2)* and su(2)* A 5u(2)* respectively. Without loss of generality we set a = -^/c and [dk x -(A, -ia) 2 coi -(|/x| [7] Constant curved minimal CR 3-spheres in CP" Comparing (3.3) with (2.14) and noting that co,, co 2 i, Po, Pi and p 2 are all real-valued 1-forms, we get Since <p is minimal, we have [5] (3.11) tr{Vco, -ipo ® co x + ipx ® co x -9~x 2 ® co} = 0, (3.12) tr{0 M ® to, + 0n ® a>} = 0 (A = 3 , . .…”