Finite Mbius Groups, Minimal Immersions of Spheres, and Moduli [Hardcover]

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.

In this book, the author traces the development of the study of spherical minimal immersions over the past 30 plus years. In trying to make this monograph accessible not just to research mathematicians but mathematics graduate students as well, the author included sizeable pieces of material from upper level undergraduate courses, additional graduate level topics such as Felix Klein?s classic treatise of the icosahedron, and a valuable selection of exercises at the end of each chapter.1 Finite Mobius Groups.- 1.1 Platonic Solids and Finite Rotation Groups.- 1.2 Rotations and M?bius 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: Kleins 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 Eigenmapslm