Abstract:
 Let $M=G/K$ be a compact Riemannian homogeneous manifold, $\lambda$ an eigenvalue of the LaplaceBeltrami operator $\triangle$ acting on $C^{\infty}(M)$,
and $H_{\lambda}\subset C^{\infty}(M)$ the corresponding eigenspace. A spherical $\lambda$eigenmap $f:M\to S_V$ into the unit sphere of a Euclidean vector space $V$
is characterized by having its components in $H_{\lambda}$. A $\lambda$eigenmap is a harmonic map of constant energy density in the sense of EellsSampson.
A conformal $\lambda$eigenmap is called a spherical minimal immersion; it is an isometric minimal immersion of $M$ into $S_V$ with respect to
$\lambda/\dim M$times the original metric on $M$. For fixed $\lambda$, the set of all $\lambda$eigenmaps can be parametrized by a compact convex body $L_{\lambda}$
in a finite dimensional $G$module $E_{\lambda}$. Similarly, the set of all isometric minimal immersions can be parametrized by a compact convex body $M_{\lambda}$, a linear slice
of $L_{\lambda}$ by a $G$submodule $F_{\lambda}$$\subset$$E_{\lambda}$. A fundamental problem is to determine the highest weights of the irreducible $G$components
of $E_{\lambda}$ and $F_{\lambda}$ which thereby give the dimensions of $L_{\lambda}$ and $M_{\lambda}$ and rigidity. Beyond this quest one aims to understand the geometry
of these moduli via Minkowskitype measures of symmetry developed for general convex sets. The aim of this talk is to define a sequence of such measures and calculate them
in specific instances. This, in particular, will give a new understanding of the "roundness" of the space of minimal $SU(2)$orbits in spheres.
