Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli

1 Finite Mobius Groups.- 1.1 Platonic Solids and Finite Rotation Groups.- 1.2 Rotations and Möbius Transformations.- 1.3 Invariant Forms.- 1.4 Minimal Immersions of the 3-sphere into Spheres.- 1.5 Minimal Imbeddings of Spherical Space Forms into Spheres.- 1.6 Additional Topic: Klein’s Theory of the Icosahedron.- 2 Moduli for Eigenmaps.- 2.1 Spherical Harmonics.- 2.2 Generalities on Eigenmaps.- 2.3 Moduli.- 2.4 Raising and Lowering the Degree.- 2.5 Exact Dimension of the Moduli ?p.- 2.6 Equivariant Imbedding of Moduli.- 2.7 Quadratic Eigenmaps in Domain Dimension Three.- 2.8 Raising the Domain Dimension.- 2.9 Additional Topic: Quadratic Eigenmaps.- 3 Moduli for Spherical Minimal Immersions.- 3.1 Conformal Eigenmaps and Moduli.- 3.2 Conformal Fields and Eigenmaps.- 3.3 Conformal Fields and Raising and Lowering the Degree.- 3.4 Exact Dimension of the Moduli ?p.- 3.5 Isotropic Minimal Immersions.- 3.6 Quartic Minimal Immersions in Domain Dimension Three.- 3.7 Additional Topic: The Inverse of ?.- 4 Lower Bounds on the Range of Spherical Minimal Immersions.- 4.1 Infinitesimal Rotations of Eigenmaps.- 4.2 Infinitesimal Rotations and the Casimir Operator.- 4.3 Infinitesimal Rotations and Degree-Raising.- 4.4 Lower Bounds for the Range Dimension, Part I.- 4.5 Lower Bounds for t he Range Dimension, Part II.- 4.6 Additional Topic: Operators.- Appendix 1. Convex Sets.- Appendix 2. Harmonic Maps and Minimal Immersions.- Appendix 3. Some Facts from the Representation Theory of the Special Orthogonal Group.- Glossary of Notations.