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Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli

"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...

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Bibliographic Details
Main Author: Toth, Gabor
Format: eBook
Language:English
Published: New York, NY Springer New York 2002, 2002
Edition:1st ed. 2002
Series:Universitext
Subjects:
Geometry, Differential
Mathematical Analysis
Analysis
Differential Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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

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