The spinor and Weierstrass representations of surfaces in space

TL;DR

This paper generalizes the classical Weierstrass representation of minimal surfaces to all conformal immersions of Riemann surfaces into 3D space, using spinor techniques to analyze the geometry.

Contribution

It introduces a new framework linking conformal immersions with spin structures and spinors, extending classical minimal surface theory.

Findings

Generalized Weierstrass representation for arbitrary conformal immersions

Established connection between immersions and spin structures

Provided a spinor-based approach to study surface immersions

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.