The spinor representation formula in 3 and 4 dimensions

TL;DR

This paper demonstrates the equivalence of two spinor-based Weierstrass representation approaches for surfaces in 3 and 4 dimensions, providing explicit correspondences and simplified proofs.

Contribution

It unifies explicit and abstract spinor representation methods, offering a clearer understanding and new proofs for surfaces in R^3, Nil_3, and R^4.

Findings

Both approaches are equivalent for surfaces in R^3, Nil_3, and R^4.

Explicit correspondence between the two methods is established.

Simpler proofs of existing theorems are derived.

Abstract

In the literature, two approaches to the Weierstrass representation formula using spinors are known, one explicit, going back to Kusner & Schmitt, and generalized by Konopelchenko and Taimanov, and one abstract due to Friedrich, Bayard, Lawn and Roth. In this article, we show that both points of view are indeed equivalent, for surfaces in R^3, Nil_3 and R^4. The correspondence between the equations of both approaches is explicitly given and as a consequence we derive alternate (and simpler) proofs of these previous theorems.