Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli

Introduction and Synopsis......Page 8
Contents......Page 16
1.1 Platonic Solids and Finite Rotation Groups......Page 18
1.2 Rotations and Mobius Transformations......Page 39
1.3 Invariant Forms......Page 55
1.4 Minimal Immersions of the 3-sphere intoSpheres......Page 67
1.5 Minimal Imbeddings of Spherical Space Formsinto Spheres......Page 76
1.6 Additional Topic: Klein's Theory of theIcosahedron......Page 83
2.1 Spherical Harmonics......Page 111
2.2 Generalities on Eigenmaps......Page 123
2.3 Moduli......Page 126
2.4 Raising and Lowering the Degree......Page 145
2.5 Exact Dimension of the Moduli £P......Page 148
2.6 Equivariant Imbedding of Moduli......Page 153
2.7 Quadratic Eigenmaps in Domain DimensionThree......Page 156
2.8 Raising the Domain Dimension......Page 165
2.9 Additional Topic: Quadratic Eigenmaps......Page 170
3.1 Conformal Eigenmaps and Moduli......Page 187
3.2 Conformal Fields and Eigenmaps......Page 196
3.3 Conformal Fields and Raising and Loweringthe Degree......Page 204
3.4 Exact Dimension of the Moduli MP......Page 209
3.5 Isotropic Minimal Immersions......Page 211
3.6 Quartic Minimal Immersions in DomainDimension Three......Page 222
3.7 Additional Topic: The Inverse of W......Page 248
4.1 Infinitesimal Rotations of Eigenmaps......Page 257
4.2 Infinitesimal Rotations and the CasimirOperator......Page 263
4.3 Infinitesimal Rotations and Degree-Raising......Page 272
4.4 Lower Bounds for the Range Dimension, Part I......Page 275
4.5 Lower Bounds for the Range Dimension,Part II......Page 283
4.6 Additional Topic: Operators......Page 291
A.I. Convex Sets......Page 299
A.2. Harmonic Maps and Minimal Immersions......Page 301
A.3. Some Facts from the Representation Theory ofthe Special Orthogonal Group......Page 307
Bibliography......Page 315
Glossary of Notations......Page 321
Index......Page 328