The Spinor Representation of Minimal Surfaces

TL;DR

This paper develops a spinor representation framework to analyze minimal surfaces with planar ends, determining their moduli spaces and introducing new families of minimal tori and Klein bottles, linking them to critical points of a Möbius invariant functional.

Contribution

It introduces a novel spinor representation approach to classify and construct minimal surfaces with planar ends, and characterizes their moduli spaces and critical points of a Möbius invariant functional.

Findings

Determined moduli spaces of planar-ended minimal spheres and real projective planes.

Constructed new families of minimal tori and Klein bottles.

Connected minimal surfaces to critical points of the Möbius invariant squared mean curvature functional.

Abstract

The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield surfaces critical for the M\"obius invariant squared mean curvature functional $W$. On the other hand, all $W\!$-critical spheres and real projective planes arise this way. Thus we determine at the same time the moduli spaces of $W\!$-critical spheres and real projective planes via the spinor representation.