Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

TL;DR

This paper classifies cohomogeneity one actions on rank one symmetric spaces, particularly quaternionic hyperbolic spaces, and constructs numerous examples of inhomogeneous isoparametric hypersurfaces with constant principal curvatures.

Contribution

It completes the classification of cohomogeneity one actions on rank one symmetric spaces and introduces many new inhomogeneous isoparametric hypersurfaces in quaternionic hyperbolic spaces.

Findings

Complete classification of cohomogeneity one actions on quaternionic hyperbolic spaces.

Construction of uncountably many inhomogeneous isoparametric hypersurfaces.

Identification of new examples with constant principal curvatures.

Abstract

We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.