The Super-minimal Surface In The Hyperquadratic Q₂

In this paper,we study superminimal surfaces in the hyperquadratic Q2.Let f:M?Q2 be a minimal(not ± holomorphic)immersion from a connected and orientable two manifold M into Q2.Two global functions ?X and ?Y of analytical type were frist introuced by Jiao to study such immersions.It is proved that in the case that both?X and ?Y are not identically zero,f is superminimal if and only if f is totally real or id:Q2?CP3 is also minimal,where id:Q2?CP3 is the standard inclusion map;in the case that both ?X??Y are identically zero,f is automatically superminimal and the normal map f?:M?Q2 is anti-holomorphic immersion isometric to f.As a consequence,we describe the totality of superminimal two-spheres in Q2.