2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152093The 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 Möbius 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.63 pages, dvi file only, earlier version is GANG preprint III.27 available via http://www.gang.umass.edu/Differential Geometrytext