Totally real minimal surfaces in the complex hyperquadrics

In this paper we study the totally real minimal surfaces in the complex hyperquadric Qn. We first give a method to construct minimal totally real surfaces in Qn from minimal surfaces in RPn, and we show that there is a linearly full totally real minimal two-sphere in Q2m with constant curvature 4/m(m+1) for any integer m. Conversely, we completely determine all the totally real conformal minimal two-spheres with constant curvature in Q2, Q3, Q4 and Q5.