Superminimal surfaces in hyperquadric Q ₂

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