Lower Bounds on the Range of Spherical Minimal Immersions

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


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 .


Lower Bound Orthonormal Basis Irreducible Component Closed Subgroup Casimir Operator 
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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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