Infinitesimal rotations of isometric minimal immersions between spheres

A fundamental problem posed by DoCarmo and Wallach is to give lower bounds for the codimension of isometric minimal immersions between round spheres. For a given domain dimension and degree, the moduli space of such immersions is a compact convex body in a representation space for a Lie group of isometries acting transitively on the domain. Infinitesimal isometric deformations of these minimal immersions give rise to a linear contraction on the moduli, and the eigenvalues of the contraction are related to the Casimir eigenvalues on the irreducible components of the ambient representation space. The study of these eigenvalues leads to new sharp lower bounds for the codimension of the immersions and gives an insight to the subtleties of the boundary of the moduli.