Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli

"Spherical soap bubbles" or spherical immersions, belong to a fast growing and fascinating area between algebra and geometry. This theory has rich interconnections with a variety of mathematical disciplines. In this book, the author traces the development of the study of spherical minimal immersions. In trying to make this monograph accessible not just to research mathematicians but also to graduate students, the author included sizeable pieces of material from upper level undergraduate courses as well as a valuable selection of exercises at the end of each chapter.

useful for a course in Riemannian geometry

Klappentext
"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. This theory has rich inteconnections with a variety of mathematical disciplines such as invariant theory, convex geometry, harmonic maps, and orthogonal multiplications. In this book, the author traces the development of the study of spherical minimal immersions over the past 30 plus years, including Takahashi's 1966 proof regarding the existence of isometric minimal immersions, DoCarmo and Wallach's study of the uniqueness of the standard minimal immersion in the seventies, and the mor recent study of the variety of spherical minimal immersions which have been obtained by the "equivariant construction" as SU(2)-orbits, first used by Mashimo in 1984 and then later by DeTurck and Ziller in 1992. 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 Kleins classic treatise of the icosahedron, and a valuable selection of exercises at the end of each chapter.

Inhalt
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.