EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 12, No. 2, pp. 551--570 (2002)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



Moduli for Spherical Maps and Minimal Immersions of Homogeneous Spaces

Gabor Toth

Gabor Toth
Department of Mathematical Sciences
Rutgers University
Camden, New Jersey, 08102

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.

Full text of the article:

Electronic fulltext finalized on: 6 May 2002. This page was last modified: 21 May 2002.

© 2002 Heldermann Verlag
© 2002 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition