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"Spherical soap bubbles", isometric minimal immersions of round spheres into round spheres, or spherical immersions for short, belong to a fast growing and fascinating area between algebra and geometry. In this accessible book, the author traces the development of the study of spherical minimal immersions over the past 30 plus years, including a valuable selection of exercises.

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.