Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces

TL;DR

This paper classifies a specific class of isoparametric functions on Damek-Ricci spaces, identifying new examples, analyzing their geometric properties, and distinguishing cases that cannot be obtained by the main classification.

Contribution

It provides a complete classification of certain rational isoparametric functions on Damek-Ricci spaces, including new examples and analysis of special cases like tubes in complex hyperbolic spaces.

Findings

Almost all known isoparametric hypersurfaces are encompassed by the classification.

New examples of isoparametric functions are identified with explicit geometric properties.

Certain tubes in complex hyperbolic spaces are shown to be level sets of specific polynomial functions.

Abstract

We classify isoparametric functions on Damek-Ricci spaces which can be written in terms of the standard coordinates $(v,z,t)$ on the half-space model as a polynomial function divided by $t$. Regular level sets of the functions in our classification encompass almost all previously known examples of isoparametric hypersufaces in Damek-Ricci space and also yield new ones. For the new examples, the focal varieties are determined and the mean curvatures of the regular level sets are expressed as a function of the distance from the focal variety. We also study the exceptional case of tubes about $\mathbb R\mathbf H^k$ in $\mathbb C\mathbf H^k$, which are isoparametric, but cannot be obtained as the level sets of any function in our classification. We show that these tubes are level sets of a polynomial function divided by $t^2$, and that analogous functions on Damek-Ricci spaces can be isoparametric only in the case of $\mathbb C\mathbf H^k$.