The spinor and Weierstrass representations of surfaces in space

Abstract

In this paper, following Sullivan, Kusner, and Schmitt, we study conformal immersions of Riemann surfaces into the three-dimensional Euclidean space. Regarding such immersions as special bundle maps from the tangent bundle of the surface to the cotangent bundle of the 2-dimensional sphere, we generalize the classical Weierstrass representation of minimal surfaces to the case of arbitrary conformal immersions. We study how such an immersion gives rise to a spin structure on the surface together with a pair of spinors and how the immersion itself can be studied by means of these spinors.