Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank

TL;DR

This paper introduces a novel construction of inhomogeneous isoparametric hypersurfaces with non-austere focal sets in symmetric spaces of non-compact type, revealing infinitely many examples in higher ranks.

Contribution

It presents the first known examples of isoparametric families with non-austere focal sets in such symmetric spaces, using a new extension method that preserves key geometric properties.

Findings

Constructed inhomogeneous isoparametric hypersurfaces with non-austere focal sets.

Established infinitely many examples in symmetric spaces of rank ≥ 4.

Developed a new extension method preserving mean curvature and isoparametricity.

Abstract

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank greater than or equal to 3. If the rank is greater than or equal to 4, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.