Recently, we have characterized together with J. Roth and P. Bayard immersions of surfaces $M$ in $\mathbb{R}^4$ by means of spinor fields, giving an spinorial analog of the Gauss–Ricci–Codazzi equations for isometric immersions. More precisely, we have shown that given a parallel spinor $\Phi$ in $\mathbb{R}^4$, its restriction to $M$ satisfies a Dirac equation $D\Phi= H\Phi$ for a Dirac operator $D$ along $M$. The difficult part lies in the converse. Given intrinsic datas: a Riemannian surface $M$, a rank 2 vector bundle $E$ on $M$, with a connection and a symmetric Evalued 2-form $B$, and additionnally a section $\Phi$ of the twisted spinor bundle $\Sigma M\otimes\Sigma E$, then $D\Phi = H \cdot\Phi$ implies (locally, i.e. on a simply connected domain) the existence of an immersion $f: M → \mathbb{R}^4$ with mean curvature $H$. In parallel, there exists a representation formula for surfaces into $\mathbb{R}^4$, known as the spinorial Weierstrass representation formula, akin to the one in $\mathbb{R}^3$, and due to Konopelchenko and Taimanov. This representation expresses any immersion as an integral over four complex valued function, satisfying a Dirac type equation. This equation was rediscovered independently by Hélein and Romon in the particular case of Lagrangian immersion. However, it remained until now somewhat unclear how these quantities were linked to spinors. In a joint work with P. Romon, we bridge the gap between these two approaches.

Recent results allowed to generalized the classical Weierstrass representation of Riemannian surfaces to pluriminimal immersions of Kaehlerian manifold of arbitrary dimension. We show that it is possible to give Weierstrass representations in the very general context of pluri-conformal complex manifolds in arbitrary signature. Using moreover para-complex forms, we find analogous result for pseudo-Riemannian manifolds of split signature (p,p) and necessary and sufficient conditions on these forms for the immersion to be (para-)Kaehler and pluri-minimal.
It is worth pointing out that this framework is particular well adapted to the study of Lorentzian surfaces, which are naturally endowed with a para-complex structure. We consider then existence results for associated families, i.e one-parameter families of (para-)Kaehlerian immersions and give some examples.