Superminimal surfaces in hyperquadric Q2

Abstract

We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions τX and τY, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f: M → Q2. In case both τX and τY are not identically zero, it is proved that f is superminimal if and only if f is totally real or io f: M → ℂP3 is also minimal, where i: Q2 → ℂP3 is the standard inclusion map. In the rest case that τX ≡ 0 or τY ≡ 0, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q2 are completely described.