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Lower Bounds on the Range of Spherical Minimal Immersions

  • Gabor Toth
Part of the Universitext book series (UTX)

Abstract

In this section, we define a new operator acting on eigenmaps, the operator of infinitesimal rotations. The name comes from the fact that this operator associates to a p-eigenmap f : S m S V another p-eigenmap \( f:{S^m} \to {S_V}_{ \otimes so\left( {m + 1} \right)*} \), where the components of \( \hat f \) are obtained by rotating infinitesimally the components of f in each coordinate plane of Rm+1. We will study the self-map on the moduli L p defined by the correspondence 〈f〉 ↦ \( \left\langle {\hat f} \right\rangle \). It turns out that this self-map is the restriction of a symmetric SO(m + 1)-module endomorphism of ɛ p that can be expressed in terms of the Casimir operator in a simple manner. In view of later applications to SU(2)-equivariant eigenmaps and for greater generality, we will define the operator of infinitesimal rotations for an arbitrary closed subgroup GSO(m + 1) acting transitively on S m .

Keywords

Lower Bound Orthonormal Basis Irreducible Component Closed Subgroup Casimir Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Gabor Toth
    • 1
  1. 1.Department of Mathematical SciencesRutgers University, CamdenCamdenUSA

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