Holomorphic structures for surfaces in Euclidean $n$-space

TL;DR

This paper extends holomorphic surface theory to higher-dimensional Euclidean spaces using Clifford algebra, introducing new representations and transformations for conformal immersions and minimal surfaces.

Contribution

It develops a Clifford algebra-based framework for conformal maps from Riemann surfaces into Euclidean spaces of dimension ≥3, generalizing quaternionic holomorphic geometry.

Findings

Derived a Weierstrass representation for higher-dimensional surfaces.

Introduced spin transform and Darboux transforms for these surfaces.

Calculated the degree of the spinor bundle for conformal immersions.

Abstract

A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux transforms, surfaces of parallel mean curvature vector, families of flat connections associated with a harmonic map from a Riemann surface to a sphere are explained. The degree of the spinor bundle associated with a conformal immersion is calculated. Analogues of a polar surface and a bipolar surface of a minimal immersion into a three-sphere are defined. They are shown to be minimal surfaces in a sphere.