Conformal geometry of isotropic curves in the complex quadric

Let $\mathbb{Q}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in $\mathbb{Q}_3$. By an isotropic curve we mean a nonconstant holomorphic map from a Riemann surface into $\mathbb{Q}_3$, null with respect to the conformal structure of $\mathbb{Q}_3$. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.