# A spinorial description of surfaces in $\mathbb{R}^4$

## Imperial College London

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.