Isoparametric hypersurfaces in Damek-Ricci spaces

TL;DR

This paper constructs numerous isoparametric hypersurfaces in Damek-Ricci spaces, characterizes those with constant principal curvatures using a new concept, and finds new symmetry actions and examples in hyperbolic spaces.

Contribution

It introduces the concept of generalized Kahler angle to characterize isoparametric hypersurfaces with constant principal curvatures in Damek-Ricci spaces, revealing their inhomogeneity.

Findings

Uncountably many isoparametric families constructed

Characterization of constant principal curvature hypersurfaces

New cohomogeneity one actions in quaternionic hyperbolic spaces

Abstract

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.