Lower Bounds on the Range of Spherical Minimal Immersions

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 R^m+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 G ⊂ SO(m + 1) acting transitively on S^m.