The Spinor Representation of Minimal Surfaces

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.Comment: 63 pages, dvi file only, earlier version is GANG preprint III.27 available via http://www.gang.umass.edu