On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

We prove that an isometric immersion of a simply connected Riemannian surface $M$ in four-dimensional Minkowski space, with given normal bundle $E$ and given mean curvature vector $\overrightarrow{H}\in \Gamma \left(E\right)$, is equivalent to a normalized spinor field $\phi \in \Gamma (\Sigma E\otimes \Sigma M)$ solution of a Dirac equation $D\phi =\overrightarrow{H}\cdot \phi$ on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán.