Moduli for Spherical Maps and Minimal Immersions of Homogeneous Spaces

Gabor Toth

Abstract: The DoCarmo-Wallach theory studies isometric minimal immersions $f\colon G/K\to S^n$ of a compact Riemannian homogeneous space $G/K$ into Euclidean $n$-spheres for various $n$. For a given domain $G/K$, the moduli space of such immersions is a compact convex body in a representation space for the Lie group $G$. In 1971 DoCarmo and Wallach gave a lower bound for the (dimension of the) moduli for $G/K=S^m$, and conjectured that the lower bound was achieved. In 1997 the author proved that this was true. The DoCarmo-Wallach conjecture has a natural generalization to all compact Riemannian homogeneous domains $G/K$. The purpose of the present paper is to show that for $G/K$ a nonspherical compact rank 1 symmetric space this generalized conjecture is false. The main technical tool is to consider spherical functions of subrepresentations of $C^{\infty}(G/K)$, express them in terms of Jacobi polynomials, and use a recent linearization formula for products of Jacobi polynomials.

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