Conformal geometry of isotropic curves in the complex quadric

Let Q3 be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in Q3. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into Q3, null with respect to the conformal structure of Q3. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.